Photon-hadron Interactions

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Photon-hadron Interactions

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Andcl-son: Dasic Notions of Condcnsctl 1lqattc1-f3hysics, ARC ppl>k, lSBN 0-201-32830-9 Atiyah: K-'fheory, AB@ ppl~k,1SBK 0-201-40792-2 Rcthc: fntcr-mctliatc Quantum Mechanics, ABC pphk, ISRK 0-201-32831- 3

Ciicmmoxtr: I3tctrocrt) namic s of Particles and Ptasrnas, ARC ppbk, 1SBX 0-20147986-9 Ilacitlscjn: Physics of Xanneutrat f31asn-ras, ABC pphk ISBK 0-201-57830- 1

ctTf-,si~agnat: Corrcrptual 1.oundations (lua~ltumMtchanics, AKC ppbk, 1SBK 0-7382-0104 9 iieynn.ran: f3hoton-Haclran fntcr-actions, ARC pphk, XSBK 0-201- 36074-8 Iscynman: Quantum l:tcctrodynamics, ABC ppbk, ISBN Q- 201 - 36075-4 iieynn.ran: Statistical ~lqcchanics,ARC pphk, XSBK 0-201- 36076-4 I:cynman: * f i r o r >of 1.utrdanrcrrtal 1'1-occsscs, ABC ppbk, fSBK 0-201- 36077-2 iiorstc.r: l4ydrotljnamic X;tuctuations, Drokcn Symmcrry, and Correlation X;unctions, ARC pphk, ISRPU' 0-201-41049-4 Ccli-hfann/Xe'eman:'fhe kightfotct Way, AB@ pplk, 1SBK 0-7382-0299-1 Gon&icd: Quantun.r Mechanics, ARC pphk, ISRN 0-201-40633-0 Kactanoff/Bajm: Qr~a~rtrim Statistical Mechanics, AKC ppbk, ISBN 0-201-41i1.2-6-X KhaIarnikoc :An Intro to thc "Theory of Stipcr-Ruitlity, AARC pphk, ISRN 0-7382-0300-9 Ma: Modern'fheory of Criticat t%henomena, ABC ppbk, ISBN Q-"138-0301-7 Migtlal: Qualitatixcl Mellttods in Quantum Theory, AARC pphk, fSI3S 0-7382-0302-9

r\lcgclc/Orfand: Quantum Many- t%articlc Systems, ABC pplhk, lSBN Q-"138-0052-2

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Q u i g : Garrgc*Ibcaricsof the Strong, Weak, and I:lccti-omagnctic fnteractiorrs, ABC pplzk, ISWK 0-0-201-32832-1 Richartison: Fxpcrin-tcntal'lkchniyues in Condensed Martcr X'hysics at I ow 'fempcratnrcs, ABC pphk lSRS 0-201 - 36078-El 12ohrlich: Classical Chargcs t%articlcs, ABC ppbk ISBN 0-201-48 300-9 Schi-iegcr: Theory of Supcrconducti~ity, ARC jq"fhk fSI3S 0-7382-0120-0 "ihwingcr: t%articlcs, "iourccs, and Ficfdr Vol. 1, I"IBC plibii ISB?; Q-"182-0053-0 Schwingcr: Particles, Sources, and Fictds Vol. 2, ABC: ppEA ISBX 0-7382 -0054-9 Schwingcr: Particles, Sctui-ccs, and Ficlcts'riol. 3, ABC pphk

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NTERACTIONS

RICHARD P. FEYNMAN late, California Institute of Technology

" L

i

/

*

~*q%

Addvnnccd Book Pmgnm,

i

- , /' A Member af the Xlcrseus Books Group

Libmry of Congress Catalogixlgin~Pu'Et1ica;tion Data Feynman, Richard PhiHips. Photon-hadron tnteracrrions I Richard Feynman. p. cm. - (Advanced book cfassics series) Originally published : Reading, Mass. : W. A. Benjamin, Advanced Book Program, 2912. (Frontiers in physics) Includes index, 1. Photon-hadron mteract-ions. 2. Hadron interacdons. 1. Title. XI, Series.

QC/93,S,IE)428F49 1989

539.1Y544~19

89-30697

XSBN 0-201-36014-8 (paperback)

Copyright O 1912, 1998

All: rights reserved. No part of this publication may be reproduced, stored in a renieval system, or transmitted, in any form or by any meatas, electronic, mechanical, photocopying, recording, or otherwise, without the prior written pemission of the publisher. Printed in the United States of America, Westview Prcss 1s a Mcmhcr of the Perscus Books Group.

Cover design by Suzanne Weiser

2345678910 First printing, February 1998

Find us on the World Wide Web at htt~:www.we~ieli~~ress.com

Editor's Foreword

Addison-Wesley's F~ontiersin Physics series has, since 1961, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics-without having to devote the time and energy required to prepare a formal review or monograph. Indeed, throughout its nearly forty-year existence, the series has emphasized informality in both style and content, as well as pedagogical clarity. Over time, it was expected that these informal accounts would be replaced by more formal counterparts-tex tbooks or monographs-as the cutting-edge topics they treated gradually became integrated into the body of physics knowledge and reader interest dwindled. However, this has not proven to be the case for a nurnber of the volumes in the series: Marry works have remained in prim on an on-demand bmis, while others have such intrinsic value that the physics community has urged us to extend their life span. The Aduanced Book Classics series has been designed to meet this demand. It will keep in print those volumes in Frontiers in Physics or its sister series, Lecturre Notes ~ n Supipkments d in Physics, that continue to provide a unique account of a topic of lasting interest. And through a sizable printing, these cIassics will he made available at a comparatively modest cost to the reader. These notes on Richard Feynman's lectures at Caltech on the topic of photon-hadron interactions, in which he developed his theory of partons, were first published some twenty-five years ago. As is the case with all of Feynman's lectures, the presentation in this work reflects his deep physical insight, the freshness and originality of his approach to understanding physics, and his overall pedagogical wizardq. As a result, this volume will always be of fundamental

importance to anyone interested in understanding the development of quantum chromodynamics (QCD)--the theory of quarks and gluons--which explains hadron-hadron interactions at high energies. David Pines Urbana, Illinois December 1991

Preface

Many of us wem first in~dtlcecito the concepts of the p m n model from this b k At h t lllme the eompting view ww one in which there were no elemenm pheles, E v e ~ p d c l e was suppsd tn, be a campsite of evey other pdcle. The ideas and conmpts in &isbook h v e helpd pave &e way far our undersmding of the constituentname of trra&ons which evennially led ts the Quantum Chmnndynamic (WD)&wq of qu&s and gluo'ns, & is m e of most of Feynmank b b s , the maximum bnefit is o b ~ n e ifd one has previowly studied the subjeer in some d e ~ fFcynmank . unique:p r s p t i v e can br;be a p ~ w i a t d by rwdws with a mlid b ~ k g o m din the subject. Although this book is dmost l8 yms otd, it still is an excellent referace, It a m a s on the raomntenkd rmding fistof &I thecmmtwD b k s , Tke bkprovidcs a g d undersmdingof ltfte m d d from rheman who invenkd it. In the "pre-QGW or "'naive" "on m d e i tfie constituents within ha&ons were assume8 u> b hunded in the transvers dlr~tion,The pmbability of finding a pmon wifftin a high momentm ha&on wi& a lmge msverrze momentum was assumed to fd1 like a Gaussian or an expnential. W D ells us that this is not exactfy &rueand givesa p w e r law fall-off in the transverg momentum, Beeause of this, many ""nve" "on mode8 e x p t a ~ o nare s mabifid (in an inzportant way) by Xogarlrhmic faem, Feynmru?used t;s laugh when his pmon mdeI was refened to as "n&ve,'"and he wollld say, ""At l m t X got it ~ g hup t to lag;&&rns." We all miss Feynman v e q much a d it is through books like this hat his id= live on,

Vita

Richard P. Feynmn Born in 1918 in Brooklyn,Richard P, Feyrrmm receivedhis %,D. from Princem in 1942, &spite his youth, he played an imwmat part in the mnhattan Reject at Las AImw World W N 11. Suwuenrfy, he taught at Gomen and at the WiEsmia ins tit^ of Twbnolqy, In 19Cj5 he r e c d v d the Nob1 Rize in Physics, along witb Sin-Imm Tanzmaga and Jutian Schwinger*far his wark in qwntum elm@dyamies. Dr. F e ~ won m his NakI Prize for suaessfuflyresolvingpmblents wirh the Wry.of qmtum elee&&ynmics, He aka e r a d a m e m a G c d of W& in~mcGonssuch fm&enral work in the man play& a key mk in the devefopment of q w k theory by puwg made1 of high energy pmmn callisian prmsses, Beyond these achievements, Dr. Rynman introducd bsic new cmputa~onal S which, Whniqws and nob~ons intophysics, aboveall, the ubiquiausFeynmn thl:way perfiaps mare than my other famdim in raent scientific hisw, have in whkh bak physiGaX qrmgses m mnceptualizd and calcufaM, was a a m a W ly effwtive &=-re Of alI his nmerous &wads,he ud of the & m ~ dM&l for Tmhlng which he wan in f 972. T k was FeynmanLechrresonPhysics, originally published in 1963.were described in ScientiftcAmesican as "tough, but nourishing and full of flavor. ARer 25 guide for -hers and for the it>est d ibeginning students." fn order to un&rstan&g of physics among the lay public, Ilr:Feynman wrote The Physicgl h w & &.E9,: Tbre Strer~geT k o v QfLighf and mat^ fir: numbr of advac& publieations that have k o m e class&refeerences an Riehard Feynmm died on F e b r u q 15, r988.

Editor's Foreward Special Preface Vita Preface

1-5

GENERAL THEORETICAL BACKGROUND Fast Order Coupling C @ ~ r n & o of n Gwent 2nd @da C~upling Uni&ty 2nd

Immpic Spin, Smgeness, CenedM C

Consemarion of C m e d M Cumn@ SJnmM#@on &@Light Cone Vacuum Expectation of V,, f l, 2) No&: h o y i n g Point Cornnzu~r

bblems

618

LOW ENERGY PHOTON REACTIONS

8-10 QUARK MODEL OF RESONANCES The Qu& Mdel Note Roblern CJcuIadon of MW Elements Feymm, Kislhge~and Ravndal, Phys. &v. (I97 X)

11-12 PSEUDQSCALAR MESON PHOTOPRODUCTION, HIGH ENERGY p%udascaiath/ifaon Phompxduetion--Nigher Energies References

13-14 E-CHANNEL EXCHANGE PHENOMENA tehannel Excbnge Pknomena Comments S-Chmnef Resomees Venezima Fomuh Estimaks af Coupling Cartsbnts

I4-%lVECTOR MESONS AND VECTOR MESON DQMIINAMCE HYPOTHESIS Roprties of Veemz Memns Elecmn RMuetion of Vecmr Mmarts Vwtw Mema Mminancie M d e l 175 as ifs VDM md Photon Ha&on In&r%tians Difhctive RMuc~onof P, M, ~t Q h r Tmts of VBM Shdowing in Nuclei To Sunnmae the Position of VDM

22-24 ELECTROMAGNETIC FORM FACTORS Elecmm;lgnedc Fom Factors Nuelmn

In b b Fmtam (emtin@ Pion Fom F=&r b t o n Fom h;ac&r for Positive q2 Note

25-26 ELlECTRON~PRQTQNSCA

DEEP mELASTXC REGION Xnelatic Eicxmn Nuclean SeaWng

fnehtic Bjecmn Nuelm Samring (eontinud) mwv of tfie Inelastic Elec&onb m n $catering

26-33 PARTON MODEL P m n Mdei n Mdef (eontinud) I'beW a Regim The Re@aaNW X = 1 The Region -q2Large Finite &gumeat &ilt ' = r Gene& R e ~ abut k the Power Law (q"f P m n s as Qwks Mamenwm C h c d by the Qwks Mdets Future Ta& of Chagd b p Inelastic Scszt&~ngwith Spin

M-35 TESTS OF THE, PARTON MODEL Angular Momentum in P m n Wave Fmcdom

-

r Expriments Testing P m n Idea (mlQ p + p v+v- + Anything EEeemn P& RducGon of

36-37 INELASTIC S C A n E R m G AS PIROPERTmS;OF OPERATORS

38

LIGHT CONE ALGEBRA

39-41 PROPERTES OF COMMUTATORS IN MQMEWIJM SPACE e o ~ d eofs Cornrnumwrs in Mommtum Spw Region I Base sr Femi Quaks Region 11 Region liIX

02-47 ELECTROMAGNETIC SELF ENERGY Efe~tFQmagne~c Self Energy Co~ghm Fomu1a Exprasion for Sdf Ewrgy in T e r n s of W Only O&cg Elw&om@e~c Energia,Qumk Mdel Elecmmpe~cSelf Mss, Q m k Mdel (continued) I = 2 2 s Differe agnetie W s Differences F d e r Cammm& on Elm CornpenEffet ~ p y +p OX Y ~ I ~n C o m p ~ nEffmt far Very SmaIE Q , V Forwad Gornpton Scattp;ringfrom Non-Relarivistic Scbwdinger Eqw~an +

4&49 OTHER TWO-CURRENT EFFECTS

50-51 HYPOTHESES IN THE PARTON MODEL

52-54 HADRON-HADRON COLLISIONS AT

EXTREME ENERGIES

55

FINAL HADRONIC STATES IN DEEP WELASTIC SCAmERINC IntPIrxtion of P m n s with the Efmmmapetie field Ragion of Finite #, V + Continuity of Large qzand Small x Region

56-57 PARTONS AS QUARKS

Preface The most advanced corn% in graduatl:theoretid physics at Caltmh is "*id Togih h T b r e ~ c aPhysics," l Esh yesu the professor &mmthe &piewith which he will deal, This y w (1971-73, having just come back from the 1991 Inemahonal Symposium on Elwmn auld Pho~onI n ~ r ~ l i oat n sHigh Energies, held at Come11Univemity, my own in&resrin the subjst was:mud, md f chase to malyu: thev&ous theoreticat questions ~ k tad hat eanference, The Eeetmsaerns1ves&me sa exmsive &at the deeision waxr made to put &em in@W k fom, with the tfioughr t hs other p p l e might atso be inmesM. Thus, the reps of the Comelt conference should be mnsidered W a compmian volume to these lectme notes. The references given here are far from comple@,but a full list of references is givert in the Rwdings of rhe Symposium, publish4 by the tory of Nuctw Stdies, Comdl tlnivenity, Jan The m & ~ dis d 4 t with on an advanced level; far ins-=, kmwEedge of the theory of hadron-hadran interactions is assumed. I have tried to analyze in detail where we stand theoretically today. The ueament is somewhat uneven; for example, I should have l&& trr study the &wryof the decay of the in more detail than I was able to do, On the o&er hand, there are long discussions of vector m a o n dominmce and af inelastic scattering. The possible consequencesof the parton model are fully discussed. Time did not pernit me to complete the original plan which was to include the Wry of weak interaction currenrs which are so closely related to elecmmagnetic cments. Many thanks must g o t Muro Cisneros whoedited,corrected,andextended the l e c ~ efrom s my class now. Wilhout his effort, this book would not have k n possible. X afso wish to &ank Ms. Helen Tuck for typing the leemre notes.

Photon-Hadron Interactions

This page intentionally left blank

Theoretica Background

Lecture l

One very powerful way of experLmentally i n v e s t i g a t i n g t h e strongly

i n t e r a c t i n g p a r t i c l e s (hadraas) fs t a look a t the@, t o probe them with a Imm p a r t i c l e ; i n p a r t i c u l a r t h e photon (no o t h e r i s E u x m a s w e l l ) .

This

per&ta a much f i n e r c o n t r o l of v a r i a b l e s , and probably decreases t h e t h e o r e t i c a l c a p f e x i t y o f trhts i n t e r a c t i o n s ,

F"or example Sn an ordinary

hadron-hadron coLlSsion l i k e np -, sp we a r e h i t t i n g two unknoms together, and f u r t h e r , we can only vary t h e energy, we cannot vary t h e

y

+p

"P

p

"t

2

of t h e pion

In f a c c a "pion f a r off its aass s h e l l " may be a

which mast be mw2. maningless

(3

- or at

l e a s t hi&fy conrplicated i d e a ,

C?n t h e o t h e r hand in.

n we knaw t h e y Is s i n g l e and d e f i n i t e , and we can vary the cgL

of t h e y by using v i r t u a l y % svi,

for e x m p l e , e l e c t r o n s c a t t e r i n g

We a r e a a s s d n e ; t h a t we do knaw t h e photon,

QED has been checked s o

c l o s e l y t h a t we know t h a t i f t h e p h o t m propagator were off by a f a c t o r of the form (1

- qZ/h2)-1

t o about 5% for q

2

then h exceeds 4 o r 5 &V.

a s high a s ( 1 &V)

s h a l l a a s w QED is exact.

2

.

The w l i t u d e s a r e k n w n

For t h e r e s t of t h i s course we

There Is already evidence, a s we s h a l l sear, t h a t

i n v i r t u a l photon-hadmn c o l l i s ions the photon a c t s nomaf Ly (i.e

., obeys

QED expectations) up t o A of 6 t o B EeV,

At any r a t e we s h a l l suppose QED exact

- where

we m a n by QED the standard

i n t e r a c t i o n theory f o r eleetrone, muGns and photons.

Exact, but i n c o q l e t e

f o r hadrons a r e charged and i n t e r a c t a l s o with the QED system. f i r s t : how we s h a l l a s s w e

-

He discuss

we can describe t h i s i n t e r a c t i o n .

Since eL is small i t is natural, t o describe t h e i r i n t e r a c t i o n i n a s e r i e s of orders i n e ,

One photon exchange, two photon exchange,

etc.

( f t might

be thought t h a t t o describe t h i s coupling we s h a l l have t o have so= dynamical theory of t h e hadrons

- ultimtely,

of course, yes

- but

detailed

some

things can be s a i d i n general r e s t r i c t i n g t h e m t x i x elements whatever the underlying hadran dynaaics

- and

i t is these r e s t r i c t i o n s we seek i n t h i s

lecture.) Tfie no coupling ease presents no problem.

The f a c t o r giving the m p l i t u d e

t h a t a hadron system goes from an incoming s t a t e In, in>, t o an outgoing s t a t e cm, out1 is: S=

-

11.1)

The S matrix 1s t h e t r a n s f o m t i o n lnatrlx fram the ""l"representation t o the "out" representation Sm

cn, i n [ = cm, out1

(1

n

*a

Tfie Beate In, Ln> m a n s a s t a t e tirt-ilch f a r i n the p a s t is a s p p t a t i c a l l y f r e e s t a b l e hadrons (otilble

in

strong

i n t e r a c t i o n s only, e.g.

no i s "stable'f)

described by momenta, and h e l i c i t i e s , a l l contained via t h e index n ,

The s t a t e

m, out has t h e s e t of indices m wgth t h e s m e space of i n d i c e s , but represents a s t a t e which i n t h e f u t u r e is asyruptotically i n s i t u a t i o n a, Thus t h e S m c r i x is r e a l l y t h e u n i t w t r r i x but i n a mixed representation,

e d i f f e r e n t l a b e l i n g f o r i n c a n g and outgoing s t a t e s .

Supposing these s t a t e s

are a l l t h e r e a r e , consemation of p r o b a b i l i t y r e q u i r e s

(In t h e s p e c i a l case t h a t the s t a t e n represents a s i n g l e s t a b l e p a r t i c l e t h e i n and out s t a t e s a r e t h e same).

General Theoretical Bmkground

The general coupling of e l e c t r o n s and hadroas is represented by t h e

The electron-photon system goes from s t a t e N t o M, t h e hadtons from n i n t o liJe suppose t h e only i n t e r a c t i o n

m out,

- and

phatm

possible is by t h e exchange of a

t h i s phaton i s characterized by a p o l a r i z a t i o a p, mamentm q:

That is t o say (supposing we could memure the amplitllde) we? define i n a given elrpertnent t h e q u a n t i t y

This is dane by renoving f r m t h e measured m p l i t u d e the knam (by QED theory) factors

I f is then our f i r s t supposition t h a t t h i s quantity

2ii(q),

depends

only on the s t a t e s m, n of t h e hadron system m d only t h e v i r t u a l =meaturn q and p o l a r i z a t i o n

of t h e v i r t u a l photon,

3,

depends i n no way an how

t h e photon was made (eg. whether by v ' s o r e l e c t r a n 8 o r on the angles and energies of t h e e l e c t r o n f o r fixed q and phaton p o l a r i z a t i o n ) . This is a s t r o n g assumption.

It has been v e r i f i e d most completely for

t h e case of proton form f a c t o r masurements, but is o f t e n a s s w d i n checking equipment, comparing r e s u l t s fro= one lab t o another e t c . We emphasize then t h a t

We assume i t .

(q) is an exgeriaentally defined q w a t i t y

definable i n p r i n c i p l e f o r all q. Me f i n d i t convtmient t o define a new matrix J non-mixed representation a s

80 we,

now w r i t e

defined i n a

-

To deal with t h i s s little more abstractly, the quantity f

QED be described a s the m r r i x e l e m n t (betwe-

U

*

%2

m, i n

9

tbe leptons) of an operator,

the vector p o t e n t i a l operator f o r the leptons s ( q ) , all considered k n m

Ft %us the f i r s t order i n t e r a c t i o n

and cempatible by QED i n any s p e c i f i c case,

is d e ~ c r i b e dby the matrix ay(q) Ju(q) d4q

+ higher

order.

The l comes f r m the zero order ( h e r e we saw the S matrix is r e a l l y the unit faaerix i n an

xed representation).

Unitarity requires t o f i r s t order

o r , @Lama (q) is a r b i t r a r y , U

U

= J

U*

-

(d

a(-q)* transpose)

J Le h a m i t i a n , Ec

Since all q are

avsflable we can define Fourier T r a n s f o m

(and f o r a lapace), Titus the couplilng f a

You m y , i f you wish simply a s e m

F" J

U M

0, c m s e m a t i o n of current f o r

hadraw, or e l s e note the following discmeion.

It is not, s t r % c t l y , true t h a t obtained from expeziment. When i t come.

from Bq. (1.5) can be c m p l e t e l y

That is because a (q) is not cmpletelry a r b i t r a v . P

f r w the usual diagram rules i t always s a t i s f i e s qUap(q)

Thus one component of a

Ft

(the one i n direction

-

0.

$1 is always miseing (unless

7

General TheareficalBackground q2 = 0) and thus one compment of J

v"

t h e component i n t h e d i r e e r i o n q

li

is

&@sing, We f i n i s h t h e deflnitiort of JP by ehoosing qI.lJU = 0. F i r s t , f o r q2

This we do i n the following way.

m

0, a f r e e proton of

-

but f o r a f r e e photon e 18 undefined t h e coupling is- e J (q) v ll bi t o i t could be added uq (e; = e + oq ) any o. This can make no e f f e c t so U Ir our consistency with QED demands cg J = O a t l e a s t when q2 = 0. This is a ll lt 2 physical property J must s a t i s f y . ZE it is not t r u e f o r general q redefine 2 q (q J )/q Evidently replace t h e o l d v i a J ' J a new J "0 Ir v U U V ' J 2 qbiJut = 0, and no new pole a t q2 O is introduced by t h e new l / q term f o r

polarization e

li

-

its

numerator qvJv vanishes e t qZ

mat

-

o f o t h e r r e s t r i c t t o n s on

.

-

0.

These we wish t o find.

J (l)?

t h a t i f badrons a r e g w e m e d

s i n c e i t is a l o c a l operator f i e l d theory suggests r-

7

by an m d e r l y i n g f i e l d theory then

l i k e aeparated (spbofizecf by

X1

For e x a w l e

(l), @

1.

2

= 0 i f 1 and 2 a r e space

you wish you may a s s m e t h i s

-

but i t is very i n t e r e s t i n g t h a t we can prove i t from our a s s a p t i o n t h a t the

s t r e n g system i n t e r a c t s with QED (subject t o s y s t e a a t i c e r r o r s i n proof due t o t a c i t assumptions.

This w y not be an i a p o r t a n t point b u t i t is f n t e r e s e i n g

s o X will. waste your clme by proving i t ) .

This i e v i a dfagrams of t h e type whose m p l i t u d e depends on a computable

f a c t o r from t h e leptons" times a matrix elelnent depending on the two a o m n t a and p o l a r i z a t i o n s of t h e N o v i r t u a l photons -1/2 VUv (g1. this

is s m e t r i c a f - i n ql

+-,

q2, u

c-t

v, f o r Bose

q2).

A8 defined

statistics a m g photons

does not p e M t us t o d i s t i n g u i s h t h e phatons s o no o t h e r function can be experimentally defined,

Using an undxed r e p r e s e n t a t i o a , and coordinate

space (via double Fourier t r a n s f o m ) we can represent t h i s m p l i t u d e as

* ~ np a r t I of t h i s course ' ~ e p t o n s fw t i l l mean s-, e

+

,-ii 4, p"

ORI,~,

-

The contribution from t h e QED lepton a i d e can be figured f o r each diagram, and c m , a s shorn i n QED theory, be generally w r i t t e n a s the matrix element (batwean i n and o u t photon-lepton s t a t e s ) of t h e tione ordered product of the operator a y ( l ) and a (2) (symbolized

Since a r b i t r a q a(l)a(Z)

IJ

be made V

Ct v

can

is a q e r i m n t a l i l y defined,

I wish now t o prove a nmber o f things, ao i t w i l l be m z e convenient

eo r e a t r i e t t h e i n t e g r a l t o tl > t 2 and w r i t e t h i s and t h e f i r s t two orders a s

Evidently one can w r i t e an e n t i r e a a r i e s af functions t o ever inereasing order.

+

This gives the r e s t r i c t i a n fusing a

+

obtained by w r i t i n g T T (L@

over a l l tl, tq.

=

a)

l and expanding t o 2nd order.

The 2nd i n t e g r a l

t2 and a p a r t t2 c tl

X t can be s p l i t i n t o a p a r t tl

in t h e l a t t e r r e l a b l e varieiblea 1, 2 (and p, v) t o g e t

NW,

i f we could a s s w e t h a t within the range of p o s e f i l e QED s t a t e s a r b i t r a r y

values for a (1) av(2) can be generated (which i s true) and a l s o independently U f o r a (2) a (1) (which is f a l s e ) we could conclude t h a t v

lii

iJ v

( l , 2) - J V ( 1 ) J v f 2 ) f o r tl

But o u t s i d e t h e l i g h t cone f o r e x m p l e a r e n o t Independent, they a r e equal,

(2.5)

t2

0 s o the

products

To proceed t h e r e f a r e more c a r e f u l l y

General Theoretical Backgroltnd wri t e

t o get

Now the f i r e t f a c t o r must vanish, f o r we could take t h e case t h a t a(21, & ( l ) c o m a r e f i r s t ( f a r example one photon from

art,

e l e c t r o n , another f r m a muan i n

I w e a t order) hence we s u r e l y always ~ u s have t

d e t e m i n i n g t h e r e a l p a r t af V

PV

In a d d i t i o n I n general we mm& have

(1, 2 ) .

(caking a d j o i n t of l a s t term)

The c m u t a t o r i s zero outsfde trhe l i g h t cone,

Xnaitrfe I believe we can mke

i t a r b i t r a w (although s o w l i t t l e f u r t h e r study of epecia2 cases i a necessary

t a v a r i f y t h i s ) hence we deduce

v y y ( l p 2)

m

J p ( l I Jy(2)

(2 99)

1 is i n foetrard light cone of 2. We have alnnost proved

i n s i d e t h e l i g h t cone of 2.

Eq. (2,5)

but not f o r a y tl > t2, anly f a r tl

The d i f f e r e n c e is very iwortanr: because

Eq, (2 , S ) , t o be r e l a t i v i s t i c a l l y i n v a r i a n t requires o u t s i d e the light: cone,

E q . (2 ,S) a l s o i a a a t u r a l if hadirons coma f r w any

rurderlying f i e l d theory, f o r then our p i c t u r e o f coupling i f Cl > t p can Lep tans

Wadrons

-

cut

be cut a t a t between t2 and t2; t h e f i r s t coupling is J ( l ) and the aecoad U

is J,(2),

so we g e t the product,

But one m y be averse t o a e s d n g t h a t

strong interactions can be deser%bed by a c m p l e t e s e t of s t a t e s (and that t h e complete @ e t can be trrlren as

[R,

in> ) a t any a r b i t r a r y tim.

Nevertheless

if we continue our study of the r a q u i r w a t e r for consistency with QED t o 4th order we can do i t , Proof

P

% i s point If

not: important, but we do include the proof f o r completeness,

(2.5) were r i g h t then 2. would be

i t is eome polyoomial in ay ( l ) , (av( (Rant. naw on we o&t the polarization $adices

way with the position indices).

- they

always go i n an obvious,

?he f i r s t order is made t o agree with Jir(l)

80 i n gemrixl w e can w r i t e CU(12) represents the devlatlon of V(l.2) J(1) J I Z ) )

f o r tr

NW &D. f o d n T g'T

fro=

t2)

t5

check u n i t a r i t y the erp - % h a factors go out.

and 41;h order i n a the U ( 1 2 3) t e r n does not enter.

so we conclude U i s hennitian aa before.

NW I n 4th order we have

TO 2nd

To second order we have

U e v c r v h e r e , and U ( 1 2)

O if

II

General Theoretical Background I n t h e l a s t i n t e g r a l we have tS of t

5

tg but no p a r t i c u l a r r e l a t i o n

t6 and t 7

and t7 there a r e 6 r e l a t i v e orders.

We replace i n each t h e verriables

tlA2, t e , t4 i n t h a t order

The l a s t t e r n gives

Uf12) U(34) a(1) a(2) a (3) a(4) : &CD

= ABCD

U(13) V(24) & ( l ) a(3) a(2) a(4) : ACBD

AF,B)D + ABCD

U(141 U(23) a ( l ) a(4) a(2) a(3) : ADSC

A C ~ ,q

U(23) U(14) a(2) a(3) a(1) a(4) : BCAD =

BE,A ~

U(24) U(l3) a(2) a ( 4 ) a ( l ) a(3)

BDAC

U(34) U(12) s(3) a ( 4 ) a ( l ) a(2)

CDAB

-

+ ABCD

+ AB@, D+ p, ~

C

+DABCD

B@,

4~ E,~

D

+C

C[D,

A]B

E, ~

D

+BACE. ]B +

C

f

+ A@,

+

ABED,4+ ABCD

B ~ D+ ABCD

(2.12)

and the f f r s c term i s

Now we g e t mny obvious r e l a t i o n s .

For example the c o a f f f e i e n t of ABCD mwt

be zero (take ease a l l 4 p o t e n t i a l s c o m u t e ) ,

We belteve t h a t the vector

potentgal a is an a r b i t r a r y functton of space and time.

We can t h e r e f o r e

chooee It t o be d i f f e r e n t fro= zero only I n four small regions of space time around t h e points 1, 2 , 3 and 4; c a l l these regions ol, a2,

03, 04.

For

o a r s p e c i a l i n t e r e s t here take the case t h a t the v a r i a b l e s have the follawtn-ng l f ght cone propertgee

CB*D] Only

D, C]

- 0

and [A,

[D,

g$0

1 is outside the l i g h t cone of 2, 3 is outside the l i g h t cone of 4.

Omitt5ng

t h e AI3CD t e r n which we noted was indepandently zero and c o l l e c t i n g what 18 l e f t i n this caee we g e t

E, ~ a11 t h e term. a r e cwf f i e i e n t s of F, AD o r CB B, 4 But vltimarely we a r e l e f t with U(24) U(13) E, AI] [B,

Only the thArd term need. t o be turned erovnd t o

C

+B E,

g E, g then

and muat a l l vanish. or ftnally

$&ace the c o m a t a t o r s a r e s u f f i c i e n t l y general, X think we can crsnclude the i n t e g r a l w i l l be zero only i f the integrand f a and U(13)

a

O even i f 1 is

outside t h e l i g h t cone of 3, End of Proof We can therefore conclude

I n the proofs of (2.9) and (2.16) we have assuned t h a t a(il. tl) and tl). a(f2,

t2g a r e s u f f i c i e n t l y general functions of 1 and 2 (when

2 is i n the l i g h t cone of 1 ) .

Me believe t h i s t o be true,

X t i s left to

those i n t e r e s t e d in Gnare rigorouhl proof@ t o v e r i f y thSe, f o r example by t r y i n g t o construct a b a s i s ,

Lecture 3

We found i n the previous l e c t u r e Vliv(l, 2) = Ju(l) Jv(2)

f o r tl

t2

General Tlzeorerical Background furthemre

This means the a r l g i n a l s m e t r i c VIIV(l, 22) can be v r l t t e n

The l a s t terns corns because t h e r e could be a 6 ( t l

a 6 4 (1-2) term o r gradients thereof--leading

-

t 2 ) t e r n , o r by r e l a t i v i t y

t o j u s t a constmt: o r polynomial

under Fourller t r a n a f o m , These s e a ~ j l lt e w would m a n t h a t t h e a b s t r a c t f a m f o r T would be l s k e

thus adding a l o c a l term a t one space time point, but second order i n a ( l ) a s appears i n QED f o r the i n t e r a c t i o n with a s c a l a r p a r t i c l e , f o r e x a m l e . Of course, i n s t e a d i t could contain g r a d i e n t s , a s F

P'J

Tktere c m be higher t e r n s f o r higher orders. t h a t T m u s t be e x p r e s s i b l e i n t h e form T

(1) F

Pv

( l ) f o r exaaple,

In s h o r t what we have found is

m

an operator &pending on a(1) only, f o r emmple of f o m 3 ( l ) a (1)

+S

Pv

l

a

l

variables,

+

,

.

P

Y

where J

,S

+

a r e operators i n t h e hadron

Such a form is of course, a l s o , t h e i m e d i a t e result: of

supposing a l o c a l f i e l d theory f o r hadrons. Research Problem % a t experiments could b e s t e s t a b l i s h existence o r non-existence of s e a g u l l s ? I n QED f o r s p i n 1 / 2 t h e r e i s no s e a g u l l , f o r s p i n 0 t h e r e l a (but with M e m r Duffin matrices t h e r e is not:

- resolve

this!)

Since a l l

q u a n t i t i e s a r e defined by experiment t h e r e a l t t y of suck s e a g u l l s f a r hadrons i s an experimentally d e t e m i n a b l e f a c t , Thus we s e e a knowledge of matrix elements of J alone shauld deternine P a l l s c a t t e r i n g amplitudes V, The r e s t r i c t i o n (3.2) an the m t r i x elements of J They lead t o nrany relations--dispersion

a r e very important,

r e l a t i o n s f o r example.

t h i s s u b j e c t l a t e r when we dlscuss deep i n e l a s t i c ep s c a t t e r i n g .

We r e t u r n t o

m e r e are

Photon-Hadron Interactions

14

eome s p e c i a l technical d i f f i c u l t i e s comfng from the highly divergent nature of some of these expressione eo mathematical r i g o r requires a l i t t l e more I n p r a c t i c e they give trouble only i n the vacuum expectation

attention.

(of, f o r example V ) and i n no other problem and eo they can be avoided beet lJ v

by disregarding them.

(They have been analyzed by Schwinger,

and a r e c a l l e d

t r i v i a l Schwinger t e r m ) . Conrervation of Current We suppose now t h a t current i e conserved i n the more conventional sense t h a t we w i l l take i t t o be t r u e t h a t quantum electrodynamics cannot determine a (1) completely, but a gradient a (1) + V ~ ( 1 )( ~ ( 1 )i e an a r b i t r a r y function, U

Li

not an operator) can be added t o i t (i.e. i f a d i f f e r e n t gauge were w e d i n the lepton theory) wlthout a l t e r i n g the physics.

- i / (1) a,, (1) d~~ = 1 - 1[ J,,(l) (av (1) + V , (X (1))) 1

Thus T

E] = TE +vX] eo

J,,

+vyx (1))

d~~

-

+/

V,,, (12) {(a,

(1)

+

(a,, (2)

o r t o f i r s t order i n X (3.5) (3.6) e t c .

Equation (3.5) we have already diecuscled; (3.6) gives rromething new.

but

(lyJ,,

(l) J v ( 2 i T

V,,{,,(l)

by equation (3.5; except t h a t V

v

-

Using (3.3)

0

does not commute with the time ordering

operation i n

C

J,, (1)

Jv(2)

0(tl-t2)

J,,(l)

Jv(2) + 0(titl)

J,, (2) J,,(l)

I T

We have a l e o t o d i f f e r e n t i a t e the 0 with respect t o t.

Thw when U = 0,

General Theoreticat Background we g e t from t h i s an e x t r a t e r a

t h e equal time c o m a t a t o r of Jo(Z) aad JV(2),

Thue i n general

I n our application t h i s leads t o

PIetually I have sfarglified the d i s c w s i o n of t h e s e a g u l l s , f o r gradients a l s o 4 appear on the 6 (l, 2 ) , but the poSnt we want t o make i s Chat the eqmk time c o m u t a t o r of

charge density

and c u r r e n t ls d e t e m i n e d by serngulls,

p a r t i c u l a r , i f as may people (@,g,

e l l - a n a l have sauggestcitd s e a g u l l s vanish

(by analogy t o QED where t h e coupling i s purely t h e r e a r e no s e a g u l l diagr-)

In

$A y $ d ~CO tr li

B

s p l n I f 2 EIald

we would have

w e have no d i r e c t t e e t of t h i a y e t kltbough we do have t e s t s of

Vbt

V

YV

(1, 2)

bmrk Sehwiager has painted out t h a t ( 3 . 8 ) is i q o a e i b l e . and Jv

-

1, 2 , 3 is v r i t t e n on the vector

3 it

Becawc: i f . Jo

says

Therefore takf ng divergence because

p

-

3

g .

But

8

is t h e operaroz Hp-pH where X i s t h e

U & l t o n i a n of t h e aystem ( a s s a n g there is one-generally s t a t e operator)

Naw

t&e

t h e energy of

OK

t h e v s e u w expactation v a l w ,

b a t t h e energy of the i n t e = d i a t e

P

0).

s t a t e n be En, we have

but

pan

-

- E.

aod i f the vacurn i s the l w e s t a t a t e we must have En

O so

wbich f s inrfrasaible, But t h i s a r g m a t applies as well t o QED i t e e l f where we Lknw we hava no eaagulls i n the oxrlg%nal f i e l d operator f o r the lagrangian, we do not r e a l l y coPrgare thire fo-1

It come because

fie2d theory d i r e c t l y t o experrimat but

rewve aom divergent vaeum d i a g r m a t the begfm%n$, l%%s problem appears

We can indeed

e n t i r e l y aesociated wfth the vacuurn problem mid could be reawed.

have the analol3y of QED; %Q seagullat a d have equation ( 3 , 8 ) s a t i s f i e d f o r ever-y problem except the vacurn,

The precise s t a t e m n t is that (3.8) holds

i f fronr the c m u t s t o r you subtract i t e vacurn expectation value t i m s r unit mtrix,

The hadronic s t a t e s n, m of a w t r i x e l e m n t of J such a@ .;m/J ir

can be c l a s s i f i e d i n t o d e f i n i t e nondyaadcal quantum n-era

U

I na

o f isospin and

strangenesa (both of which, we a c l s w , a r e perfectly consemad by the strongly intaracting syatem above).

The J nratrix say have elemnta between different

values of the quantm riders, but: i t mmt of Course coa@erve charge. F r m the very low race of K'

-*

no

+y

er d

+

y

+n

(alt;bou& C@

+

East enaugfi) we cmclude Cltat weak interactions a r e invalved here, think

J

n

+y

#.e

Thus we

has zero nzatrix elements bemean etaces of different strangeness,

h m g a e t s of d i f f e r e n t isoepin we can descrlbe the r e s u l t by saying J has

p a r t s of X spin

fn

0 (isoscalar), f:

I (isovector), 1

2 (i~aoteaaor)etc.

and w e appropriate Glt?bsch Gordtvr coefficianta t o r e l a t e ampxitudes amng

diffetctnt multiplets.

Slnee J daee riot change charge i t nnust only LnvoLve

the 3rd emponeat, i f it: i e isovector for e x a ~ p l e , The f a c t that proton a d

General 'rheoreriealBackground

+

neutron have charges

l, O already shows that J Is not pure i s o s c a l a r

independent of isoapln, nor pure isovectox (where the charges would have t o be opposite) but contains a l i n e a r co&inatIon of these two, t o require X tested.

No expsrixnent s e e m

2 , but I do not know him precisely o r extensively t h i s ha5 been

Recently so= evidence w a s claimed f o r the need of an 5

i n comparing y p

+

n;'~

and y N

i .

2 coqorxent

%-p a t energies near the d resonance--but i t

appears that correctiaas f o r dauterm s t r u c t u r e (for The y neutron r a t e is inferred fram y D data) *re

incorrectly analyzed.

Host t h e o r i s t s today a s a a e h1

.p

O o r &X:

P

L only f o r J , (This i s 1.1

evidently s fumdamntaf. question because i t t e l l s somthing of haw 3 is " u l t t m t e l y '"coupled

; f o r further strong interactions c m s e r e n g X-spin

cannot a l t e r & i s rule-+@ see ""ln'>tbroulgtr

the strong d p m i c couplling i n

t h i s respect a t Least because the strong coupling consemas t h i s Z spin eharactar ,) Having available a ~ a t r i xelements cmlJ in> f o r a variety of s t a t e s n (and m)

v

a l l s f the s a m X-@pin m l t l p l e t

pe&t;s

an@by appropriate l i n e a r cmbiaation

always t o i s o l a t e the pieces due t o the i s o s c a l a r and isavector p a r t separately.

Thus we can define matrix e l e m n t a and therefore operators f a r J5 (q) nnd J " ( ~ ) . ?J

U

But f o r the vector we could a l s o calculate (via Cl&sch Cordm ccefficients) m t r i x elements b e t ~ e o s! p ~ e c i f i c s t a t e s of other cmponents of the vector current jW(g) o r

(with isospin + l o r -1).

U;

I n t h i s way new kinds of

currencsare defiaable. "Illis would J u s t be an exercise i n Cleibech Clcrdan c o e f f i c i e n t s , but we think so= of these currants a r e p h y e i c a w important also, current ~''(~1 is the

v

W&

interaction

Ffe think the

nonatrangenaes e h a ~ g i n gnonparity violating parr of

(an a s e w t i o n k n m ss WC). This leads t o a suggestian

by which these extensions gf current: a r e w e f u l i n a p w e r f u l t h e o r e t i c a l w a y (&ll-Wn) * is (in our point Ft of view i n these lectureslakind of accident i r r e l e v a n t t o strong interactions,

That QED i a coupled t o JU, o r we& interaction

t o+'J

They J u s t lead ucs t o a t o o l t o study h a d r w , but: hadroa &ateractions can be analyzed alone.

Nevertheleas we deduced a. rider of r e l a t i o n s from asfj&ng

e i t h e r t h a t these currents c a m from soae apeastor i n a f i e l d theory underlying

hadrons, o r t h a t hadrona a r e suck that weak perturbation f i e l d s could be coupled (we w i l l w e the l a t t e r hypothesis),

We can expect, f o r any two points and

components of current ~ ~ ( and 1 ) ~ t ( 2 )that they c o m t e i f 1, 2 are apace P Like separa ted

llhis La a new arsrtvtion, on J and the hadron system.

We a r e trying to Lnduce new laws md restrictLonrs

We know

V

- 0 for 1 x . 1

our electrowgnetic current, is an Lsovector and isoacalar,

where J,

Ustag isospin only

how f a r is i t possible t o go t o prove say t h a t the tsovector p a r t , o r the

+

etc. c m a t e l

Xn the realm of isospia what we ass-

here 1s that

the apace-like carnutation law is t r u e not only f o r the t o t a l curreat J~

+ Jv3 but

ale0 f o r the isoscalar part alana with i t s e l f , the isoveeror

part alone with i t s e l f , the isoscalar part: with the iaovector part and generalizations fox the Leovector p a r t with different I spin compment directioas. We caa a l s o have the s c a t t e r i n g of an 3larogPnary a-type vector "photont' t o a b-type photon governed by

That is, extend the concept of vector potential a (1) t o contain mather U

Lndex a , the type, carrying f o r e-ple

isospia o r straageness,

men the coupling

t o external potentlals could be

Gonservation of h n e r a l i z a d Currents

-,

We want the consewation laws analogouss t o V1.I V v*' (1 2)

-

the case i f I spin which we k n w is exactly conserned. of an I

l p a r t i c l e v i a JV'. U

than the i n i t i a l , so

m

0.

Take f i r s t

Consider the scattering

?he charge of the f i n a l s t a t e is one higher

General Theoretical Ba~kground

;3

(tl,

I),

V+ JP (tl,

Xf

(4.51

but t h e last, equal tiw c o m a t a t o r is by hypothesis (4.1) zero outside the 3

mwt be a m u l t i p l e of 6 (g-g'),

l i g h t cone-it

V+

shows s must be J y (g)

.

say s ( ~ ) 6 ~ ( ~ - x " )Equation . (4.1)

Hence t h e equal time commutation r e l a t i o n r e s u l t s ,

( t h i s a s s m e s t h e r e a r e no s p e c i a l t e r n i n

8(x1,

E2) &ich would i n t e g r a t e

o u t , a question we s h a l l s e e r e l a t e d t o s e a g u l b again). Eqtxatlon (4.6) and its g e n e r a l i z a t i m t o t h e much wider group SU3 x SU3

arc! Gelf-&nnfs equal tim comulirtor r e l a t i o n s ,

They represent the f i r s t

guessed d p m i c a l property of hadrons t h a t is nor simply a consequence of r e l a t i v i s t i c quantum mechanics geaeral principles, We c m a l s o describe t h i s from t h e point of view of a property of t h e s c a t t e r i n g function . : ;V

Talre the case one p o t e n t i a l is

+ isospio,

o t h e r I$. Since current is conserved you might e t f i r s t think Pp, ;V but t h e e r r o r is t h a t t h e p o t e n t i a l v, 2 c a r r i e s i n a charge e l e c t r o m w e t l c p o t e n t i a l (coupling 3) a180 conserved,

C O U ~ ~t o~ the B

+.

the (1, 2)

So only

+ rneson

tyi

-

the

i s charm

E.g. d i a g r a m a r e l i k e

Their sum eonservea charge, and ift h e photon had p o l a r i z a t i o n e proportional t o i t s oun momentum t h e sum would give zero.

y. =

~ ~ ' ( 1 2, ) is j u s t )J v

the s m of t h e f P r s t two, t h e l a s t one is (38~~flly cmputcible and is c l e a r l y a f i r s t order hadronic nracrix e l e a e n t of a current J

v'

i n thLs cage J~ i t s e l f .

musr one em e a s i l y show t h a t

o~J:(~I

AS

-

3 ( j u s t by I-spin from V J (1) Er Er

we shall, show i n a m m n t ( 4 . 7 ) is equivalent

CO

o

(4.6) i f no s e a g u l l s e x i s t ;

i f they do e x i s t ( 4 . 7 ) 18 t r u e but (S. 6 ) has t o be madifled.

(4.7) is t h e

0,

more fundamntal r e l a t i o n , t o (4.6) we s u b s t i t u t e (4.3) i n t o ( 6 . 7 ) -

To show t h e r e l a t i o n of (4.7)

s o w r i t i n g 64(1-2) on the r i g h t s i d e of (4.7) o r 6 (tl-t2) the r e s u l t (4.6) s i n c e

g

,J3 = 0. U U

63(i1-;21

we obtain

I n general s e a g u l l terms must be added, b u t

we do not k n w i f they e x i s t . The generalization t o a general Lie group with generators commutation r e l a t i o n [ G ~ , $3

V~,V;;(~,

= fzb

(

with

by d e f i n i t i o n ) (wb), = fzb

( f r . 10)

2) = 64(1-2) 3 Y b ( l )

m e s e m y be obtained from noting t h a t the generalization of a gauge t r a n a f o m t i o n -r a + P X is, i n the group aa -t aa + qVXa + (X P P D P li unchanged by such a t r a n s f o m t i o n we Efnd

a'

a s a f u n c t i o n a l relation-or

X

ay)'.

Supposing T[d

is

c a l l f n g 6T/Sa(l) t h e functional d e r i v a t i v e one

e a s i 1y deduces

f o r afll X ( l ) , s o i n t e g r a t i n g by p a r t s we have

kthen T is w r i t t e n i n a p w e r s e r i e s i n a (4.2) and is s u b s t i t u t e d i n t o (4.111, zero and f i r s t order t e r n give (4.9) and (4.10). Since idsospin is exactly cclinserved (4.91, when SU

3

(4.16) mwt be e x a c t l y s a t i s f i e d

r e s t r i c t e d t a trhe t h r e e s p i n components of vector c u r r e n t s ,

% a t of

which i a only ""almost" s a t i s f i e d ? G11-Mnn has proposed t h a t $U3,

although

nor: exactly s a t i s f i e d f o r the e n t i r e hadronic system may be more and w x e

accurately s a t i s f i e d as shorter aad shorter space-tim i n t e r n a l s a r e invalved, That is how i t would behave i f an wderlying f i e l d theory had propagator gradient terms s a t i s f y i n g SU3, but mass-like terms violating SUy

+ Gmq

where m is a nonSU3 invariant matrix, q a r e quark operators).

is a strangeaeas changing curreat, h a d n g f o r e-plf? A, N then v .Ja(l) $ O becwse A, ~t v

= O, < A / JEI E>

-

If

J'

a w t r i x e l e w n t beWeen

have dif ferent mases.

a ~ z u say, therefore +allE 1 IN>

cannot be zero, ) That: is P $(l) P P na(2j. Then

9 q+

(E .g.

(If A, N a t r e e t

- = (uA-wHjo

fs equivalent t o another operator, say

4 H c r v the l a t t e r p r e s w b l y does not contain a s i n g u l a r i t y a s strong a@ 6 (1-2)

us say the 64 (1-2) s i n g u l a r i t y

but i f SUg is v a l i d a t small enough distmces-let of V vBb is correctly given by (4.10). Ir Ctv

%vs we say

4 where "ssfaootk" is l e s s singufar than 6 (1-2).

eqttaf

tinrc?

Then we c m s t i l l deduce the

e m a t a t o r r e l a t i o n s under the above a s s q t i o n s of smoothaees of

the SU3 vioXating term.

Equating the singular t e r n i n ((412) aad (4.12)

we f i n d

(eeagufl t e r n have been imored). These relatfons a r e 05 very great I n t e r e s t because they a r e aonliaear requiring absolute scales,

Thus ( i f valid) they can serve as supplying

absolute s c a l e definitions t o the currents so t h a t the r u l e t h a t weak i n t e r a c t l m of hadrons i s V therefore testable,

+A

(rather than V

-

Thls p a r t i c u l a r t e s t has bean m d e by Adler and

Weisberger uslng fCAC t o take the pion coupling of the a x i a l current.

.7A) is d e f l n d l e m d

We d i s c w s how @-**at

ah,

a mmure? of the divergence

m r e d i r e c t t e s t s c m be made

by neutrino scattering l a t e r on i n the course ( i n Part XI),

I s zero i f 1 is outeide the l f g h t cone of

2 , nonzero

inside.

What kind of s i n g u l a r i t y does i t have j u s t passing

through, o r near t h e li&t cone,

For f r e e f i e l d s of any noass i n f i e l d 2 theory the c m u t a t o r has a 6(sI2) type s i n g u l a r i t y across t h e cone. There

i8

experimental evidence from i a e X m t i c s c a t t e r i n g experilnents of e l e c t r o n s

on protons t h a t the s i n g u l a r i t y of

Ir

Is of t h e s m e type

2 (gradients of) 6(s12) where s12 is t h e i n t e r v a l from l t o 2.

We s h a l l discuss

t h t s FBatter i n coasidttrable d e t a i l where we discuss these experiments,

At this

time we s h a l l a l s o develop f u r t h e r o t h e r formal properties of the c a r n u t a t o r o r tlm ordered products of J

li

operators,

m a s e matters w i l l therefore be

deferred u n t i l l a t e r , f o r I f i n d them e a s i e r t o d i ~ c u a swhen c e r t a i n experiments a r e i n &nd,

I n order, however, not t o leave t h i s t h e o r e t i c a l diseusslan eentirely i n t h e a i r , I s h a l l i l l u s t r a t e one a p p l i c a t i o n of it--the simplest, t h e vacum expectation of V

I.lv

( l , 2).

ThSs is a function only of t h e d i f f e r e n c e l

- 2,

Its Eouxier t r a m f o m , i n t o v a r i a b l e q - d i c h we w i l l c a l l Vpy(q) La needed, 2 f o r e x m p l e , t o c a l c u l a t e vacum p o l a r i z a t i o n correctione ( t o order e ) due to hadrm.

It represents t h e d i a g r m bdrons

If we w r i t e i n momentvm space 4 [ V (-q, q) /o> = -q q v(q2) Ir v

Ft v

+

2 6y,b(q ) (by

2 r e l a t i v i s t i c iavariance, gauge invariance implies b = q v o r we have 4

1 vp, (-S,

4) / 0'

I f we w r i t e ( b y v a new pole a t q2

(6

uv qZ

- 24p9v ]b

-

- qvqv) v(q21. we s e e b mmt go t o zero BS q

(4.15) +

0 i n order t o avoid

Q

O.

Acting on conserved c u r r e n t s t h e l a s t term vanishes.

TIse serge@ of buibbles propagating between two c u r r e n t s is

23

General Theoretical Backgragnd We note there is no mass r e a o m l i z a t i o n of the photon, the pole is s t i l l a t qZ

-

0, but the residue i a altered t o

4mL where a v(O) (vhich -----y

1-4ne i a t u r n cuc i n f i n i t e ) is lost i n charge r e n o m l i z a t i a a , 2

near q2

a

O t o get v(q )

t o f i r s t order i n e

2

+

my

Tf we s t a r t expandlrng

2 q b, we can write the renormalized propagator

as

so 4ne2i b measures the vacuum polarization correction due t o hadross i n such predminantly low enerll;y QED problem as the h& Shift etc. 2

I h e imeginary part of v(q ) f o r q2

O is the "virtual photon ltfetima"

ltnd gives the r a t e o f production o f hadrons i n (say) an electron-position eolliaion,

Because the iwginary part of the mplitude represents a loss i n

probability that a photon remine a photon

+

m

l

+ 4 ~ e ~ & vi.e. ( ~ ~ Prob. )

1 + 4ne2 i (v-v*)

X t is therefore dgreetly accessible t o experiment,

to tbe imginary p a r t by a dispersion relntion.

2

2 + e-

lae i ,p1)7 'moa, / '$ (4)10' 9

detemined a f t e r suitable

We w i l l discuss t h i s matter i n d e t a i l i n the aexr Lecture.

Consider the process

iE2y

The r e a l part is related

Therefore hadronic vacurn

polarization e f f e c t s (to ordere2) could be c-letely experimnts a r e done,

.

+

hadrons iln s t a t e ra out,

It is gmeretad

Hence t h e p r o b a b i l i t y i s proportional t o

Thus i f we could Bleasure t h e t o t a l cross s e c t i o n f o r e*

+

+

hadrons i n any

s t a t e as a function of the energJT E of t h e e l e c t r o n o r p o s i t i o n i n t h e case (qo = ZE,

m out

6

'01

-

0 , qZ = 4 ~ ' ) we can d i r e c t l y measure Zmoyt

Imout>(mau,/

JP(-%)

t o w r i t e t h i s i n t h e Eom

W

- q26 UV ) 8 (qo)p(q2) because we know t h a t

being lowest) no hadron s t a t e mout

$I

o r q2

. -

q2 p(p

?-

2

.

(2mV12 and q* 2

Par erample, p(q )

0.1

-

O f o r qo

0.

e

PV

(g)

-

Q ales8

~ h u sp(q2) exist. only f o r

O f o r q2 < 0, i.e.,

Is i f q

mwt be p o s i t i v e , f o r e

puV(q)

is nonzero, t h e lowest possible

s t a t e is a p a i r of pions with mmntmn Q hence p

2 (ZmT)

O ( t h e vacum s t a t e

if qo

could be excgtad.

(Xn f a c t i f Q, t h e apace l i k e m e n e r n of

qo > 2

/ Jy (q) 10,

~ ~ ( / 042 )

R e l a t i v i t y and gauge Savariance p e d t s (quqv

0 ' / Ju ( - g ) / moyt> (2%)

2

,

.

Therefore

(F, T, s t m b f o r Fourier trmsfanen),

NW we can work out the c o w t a t o r md tim ordered product,

Note t h f s c m be m i t t e n as

F i r s t the comatator

25

General Theoverr'calBackground Therefore the vacurn expectation of the c m u t a t o r i n space is

which i a zero outslde the Light cone, becawe ~ ~ ( 1 , 2is. )

It is i n t e r e s t i n g

t h a t the proof t h a t the esmutaeor vanishers outside the l i g h t cone, a s a d n g r e l a t i v i t y is s o skmple for t h e vacuusn ex;pecCatian.

(At the end of these notes

the e o m t a t a r , propagator and t h e i r Fourier t r s a s f s m s a r e I t s t e d )

.

To get

the time? ordered product we need

To gat the F. T. we need a convolution with the F. T, of @(tl-tZ)

Therefore

P* T.

CO/ i

10%

where q' meaoa ( q i , 'Q = qtqf

-h

~ ~ ( (211 1 ) ~ ~ T

Q).

F l r s t do the case li.v

- 6 t t q2 and comes o u t of

-

tst

the integral.

for d e n

q'q'

Ir v

- Spvq' 2

Then change the s i g n of

q f f%nthe second t e r n i n t e g r a l to get

-

F.T. C O /(

J~(~)J~(~)]/Q5 , (qtqt

- ,e

dp; O

-

- btt9 2)

V t t (q9

The i n t e g r a l i e just l

'

qo-n'2+i.. 0

ar chenging vargnble Vt,(P)

= (ntqt

from q; t o q;2

- 6t,P29

- qZ

-?

get

1

(5 e 9 )

&are Set 2s the F. 5, of' a gosssble s e a g u l l a w c t a t f o n i n the vacuuat, m e t be

8

c o m t a n t , or a fZnIte polynoeaf. i a q, and muet be r e a l .

It

W can

w e s s (generalizing wv from tt--however s e e note belrrw) t h a t writing VyV(q)

(qiiqy

- 6Uv421

v(q2) we have,

where S, a t worst, is a E t n i t e polynoaalal i n q; a t b e s t f t is zero, imaginary p a r t of i v ( q2

1

Note the 2 p14 1 a s previously remarked physically we can a l s o

write

This is a dispersion r e l a t i o n f o r v(q 2 1, expressing v f o r a l l q 2 i n t e r n of its imaginary p a r t (which i n t h i s -%se is (a) nonzero only f o r q2

2 (2mw) , (b)

m a s u r a b l e by accessible experiments.) I f ( ~ ( s )is the cross s e c t i o n f o r a n n i h i l a t i o n of electron-positron f n t o m y hadrons where s is t h e t o t a l C.m. energy squared, we have

a(s)

(4~e~)~p(s)/2

s o the vacurn polat.Zzation is given d i r e c t l y i n t e r n s o f experiment bp

The Lamb s h i f t correction (or eorrection t o the magnetic m m n t of t h e e l e c t r m , 2 e t c . ) due t o hadion vacum polarf zation depends on 4ne i v ' ( 0 ) which is

X t is expected today fox reasons t o be discussed l a t e r t h a t o ( s f m y , f o r

large

S

vary a s constant/s sa the i n t e g r a l would converge and be d e t a d a e d

ewerimatally. Xn general t h e possibility of use o f such

B

relation.

2 value of q (SW O o r

m)

;uz

S mwt be resolved t o get trhe g r e a t e s t

otherwise i f v(qL) is k r ) m sawwhere

a t some s p e c i a l

we can convert t o a subtracted dfspersion r e l a t i o n .

I.e. suppose f t i s k n m S Ss a constant, but n o t a p o l y n o d a l i n,'q 2 is k n m . -#(g1)

Subtract from (5.10)

S does not appear.

f o r q 2 its value when q2

and

2 ql t o get

This s m e t r i c k helps i f the i n t e g r a l on mL looks divergent.

Cer~eralTheorerim/ Bmkgr~und Ihe integral on m2 may n w converge b e t t e r f o r large m

2

.

In our application any conetant S i e uninteresting, as we have seen the value of v(O) is of no i n t e r e s t today became it: is l o s t i n shazge r e a o m l i z a t i o n , Hence supposing S does not have a q2 term (aesumptioo of no bad aeagulle) we would write a dispereion relatLon f o r the quantity v(q2)

- v(@), the only quantity

of physical significancct

Thua we m y look fomard i n the near future (when e x p e r i w t a l r e s u l t s f o r

2 p(m ) w i l l be =re

2 c o w l e t e ) t o being able t o W e a ( f i r s t order i n e 1

correction i n QED calculations of the e f f e c t s of loops invo1Mng hadrons,

+ * p+v-)

In a recent mwkturemnt of ~ { e-e

2 of the Q1 resanance t o p (m ) has been observed,

an e f f e c t o f the contribution n I s cows from the interference

of the graphs 4.

It

+ 2

2 a t a q near aa4,

S, .

It

m ,

e

e

In the first: order 5n *%eh the e f f e c t contributes only the

Is seen, t h i s is observed as a s l i g h t o s c i l l a t i a n

AbS an e x m l e we cctlculate the rnodfficatlon to @(@'em-t. y * ~ " ) when '"adrons'"

i n the above graphs is the $ resonmca only,

Zn this ease

CS, 1s)

(S. 17)

Mhere p mans principal part,

28

Photon-Hadxon Interactions l m e r l i m i t i n t h e i n t e g r a l (5.18) can be s e t t o

e r r o r because t h e resonance is very narrow.

The cross s e c t i o n f o r

-"

with n e g l i g i b l e

Carrying out t h e i n t e g r a l we find:

u+p- production from ef e- is t h e r e f o r e

where a. is the c r o s s s e c t i o n f o r the process i n the absence of @-photon interactions. We need not have evaluated t h e Pntegral i n (5.18) t o g e t the r e a l p a r t 2 of iv(q )

.

Uhat t h e r e l a t i o n (5.15) says is t h a t iv(qZ) is an a n a l y t i c

function with no poles i n the l m e r half plane and such t h a t Zm a l l we need t o do is guess an a n a l y t i c f i m g l n a r y p a r t is

.l 2

T

p(=

f and which has the c o r r e c t poles i n t h e upper h a l f

Xn t h i s case, t h e function is easy t o gueas

plane.

theref o r e

Meking t h e s u b t r a c t i o n a t q2 = O we get (5.19).

Note: Let us calculaCe o t h e r uu components in (5.8). P;

F i r s t t n s o q;q:

- bpVqtZ

QX*

m e r e f o r e get

80 this is OK b e c w s e t h e f a c t o r q0 t t case,

q:

-

Qx

m l t i p l i i e s the serne i n t e g r a l a s i n t h e

Traubla come h m e v e r i n the m c a e f o r then 2

2

( q i -Q 1 f o r the q;2

the c a e f f l c i e a t i s

does n o t change d i r e c t l y t o q2 a s required, but

we have i n s t e a d an e x t r a correction of q i 2

- qoZ which

cancels the denominator

80 we get

&ere C i e a constaint ( i n f i n i t e no doubt)

5

We ewld get r i d af C by a s a a g d l type t e r n but we a r e eoafused becauee ordered produet alone s e e m not t o be t e l a t i v i s t i c a l l y i n v a r i m t ,

the t*

Apparently (*l2

f o r q2

m2

-

+

Evidently we a r e measuring

0. me ( t o t a l energy io the

2mvW

where vLAg f s the energr

c f the pihoton i n tbe lab system (Ear us v goes Eroa O t o 2 say),

nat the m l y atcltrix e l e m n t s from I s t a t e s a t these enotrgie8

etc. a r e a l s o possltble but. we d%aews these l a t e r ,

mese a r e

The

rullst

&exacteristic fearure of the behavior Fe rapid chmges i n mgular

d i s t r i b u t i m with enerw.

su&

m a-

+p

+p

-+

r"lAs feature ilr c h a r a c t e r i s t i c of hsdrm c ~ l l i @ i o n s

alao a d is Snterpr&ted aer due t o resonanca.

s m e z e s w w c e s appear here, of

COUXE~, W

i n s scatteriag,

The

The ghafon data

has been emlalaed ancl rrnalyzed i n consfde;c&la derail. on t h b b a g s ,

F%mt

we d i a c w s tbe general ""eeory" oof tesonrsncesrr. of the T guarr%xmwt have a apecisl propatty o f

A g m e r a l e1-at:

separability def Sned by tbe f o l l w i n g circcolllsian like G

+X

t o wh D

J.

B

+G

J.

D

+E

+ P.

taprces,

Suppose we img&ne a

h a pasaibiliey is t h a t A

in one part of space (via e l e m n t

C ~ I S I A Bm>ci then e B

r e e l p a r t i c l e goer off acrosa the. roar5 t o hit: G t o d That b

~ E E T ~ I must T / ~have > m infinity--a

eatiafy p*

.%7

+ pg

- pc

pX and :p

% is positive)

hit first

X W

8

+ F via G + X

+

E + F.

singularity when the maenta

hru Ule valve

%2

f o r e r e a l pareiela X.

residua a f the singvlarity .i ~ R / T \ ~ N ~

f o r each resonance had t o be deternrined e w i z a s a l l y , by ad;justAnlng t o f f t: data, (The other f a c t o r < E N ~ T ~ Rw>a s ssv@jll&le from n acacterisg). ( 3 ) The '%ackgraundv' was a slawly vatyZng -litode

i n each chameX,

CootSnesai. of f i t of paramfaro i n (l) i e r judged i f the background can be made t o vary tslauly, o r at; l e s s t s m o t h l y ,

backgrqround t e r n would be r e a l but:

T t was hctped t h a t a11 theae

smU imaginary nm ultimately

needed i n so= c h a m l s , possibly beeawe resonanees a r e left out, or have

37

Law Energy Pltororz Rcactr'orrs fnaccurate p a r w e t e r s ,

mext time

Walker does t h i s he w i l l r e l a x the condition

t h a t the background m p l i t u d e be r e a l , )

Lecture 8

We continue with the di~sua~a~ilon of the t e r n used by Walker.

(2)

t c f i m e l a pole,

T h l e Is fsm diagrslcn A

la diagram which does not e x i s t f o r +@--but r e %Illsay aore about t h i s l a t e r ) . E t leads t o s term l i k e

-

where

8

is an m p i r i e a l determination of how a ' s couple a t t h e i r

pole, empirleelly

This f a c t o r t-42 varies rapidly with t f o r

14.8.

eaaall t , lesdtng t o a rap%&angular dependence. may

from the nucleon.

S m l 1 t lneaasr

lan

effect far

m e e f f e c t 18 t h a t s nucleon hais a chance t o be seen

ss a nucleon with a n' around i t spread r e l a t i v e l y f a r out, a s

i n mplitude.

I f thh%s v i r t u a l

B

is k i t with a photon f t ean be e m t out

fomards aad. appear r e a l ( " d o a d " ' ) having received its necessarg energy and meat=

f r m the photon,

It goes f a i r l y aharpLy forward--at

higher eaergiee

a t l e a s t and a f f e c t s the m p l i t u d a s fn a l l valuw of s-&aanel lrtaplar mmntraaa incluBing high v a l u w , where f o r low energies and =mat& no ferraaaaces a r e large,

So it gives the nrajox c o n t r ~ u t ? l a aa t tha h i e e r angular

Since there the rapid v a r i a t i o n wgth t near t pale 42

.OZ

4.:

0 f.s essential

we;

a r e near the

G ~ V ' and we can e m i d e r It trueororthy and accurate.

To be

sure, a t the higher energiee lawer sagular mmmts invoxve i t f o r t; l a r g e and

SO

the f o r .

l/t-Getc. may be wrong o r smblguow (we

follow the principle--the

man we r a n t t o

contr%butioa of poles 2s well defined only f o r

pm-ters

near t h e i r resonaace),

But here the a m l i t u d e is slawly energy

d e p d e n t and i t s exact value Ear the l m a s t one o r

WO

angular m e n t a

(a, p wave@) is l o s t %a the background mplltudea we a r e rrddgng &away.

(He backwomd was added f o r very high angulsr =mattan). The exprearsion we have written f o r the a p l f t u d e i s not gauge invariant, hwwer,

f

The R springs lrm a source m4 doesn't conserne charge,

be co:orablned t o other t e m o t o &a

f t cowletesly gauge invariant.

T t arust

Qne obvious

c a n t e a e r f e the a channel resonance cosresponding t o the unexeited nucleon a t 938,

We

be e l f i i n a l

C=

pat these together asc follcrws ( l e t chargas on Initial. nucleon

nuclem e2, pion en)

these &&v@

With

4

replaced by

the three a q r e @ a i o n sgive

Note %a i n e l w i a of d i a g r m I Z T f o r the nuclean reeonance r a i a d e ue of Eh@question of whet;her we should not: a l s o have added to our law energy s channel

Low E~ergyPhoton Reacti;ons resonances, such

W

the 4, such d i a g r w as

The w w e r i s certainly yes,

They were not included by Glialker but again they

are r e l a t i v e l y s l w l y varying f o r a positlve e n e r m w ,

They a r e f a r off from

their x e s o m c e (hi&appears belw thxeahofd at negat5ve The gauge invariance of Mx

+ MIX +

U$

is seen t o hold even i f tbe

r t e r n a t e d t t e d , m e e e t e m contrgbfbute only low sngular

an-loa

momentm a t law energy as does the e n t i r e I T and Tlf, and does not vary p a r t i c u l a r l y rapZtlly with k. l / k singular behavier. constirnt

ets

k

-t

For small L a11 three t e r n have the expectad

The mamlausl m g a e t i c te-

O beeawe there

&B

l n I1 aad TIE go t a a

a k In the n w r a t o r also.

Since the?.

b a c k g r o d t e r n was a r b i t r a r y except it was rusrsmed t h a t it was a) smll i n hi&er angular mementm s t a t e s , b) slowly vazylng I n energy; we could omit the a m l o u s mmeat t e r n asd inelude thea i n the background,

Walker did t h i s

and found, perhaps surprisingly, t h a t Sf be l e f t these anmZous mment tern

out, the backgroud

watit

d e f i n i t i o n of the '"pfn

a a l l e r than i f ha l e f t them in, exchange tenn" is I

his e w l i c i t

%W

+ IX + 111 with v1

p2

O

(an e w r w a i o n he c a l l 8 the?. ""elecrxie Born tern"'), We Bee t h a t he caulit j u s t have w e l l l e f t out the nomon;taloufl w m n t t e r n too d w e d a s h p 1 e r exprearsion--readily generalized t o etake gauge i n v ~ r i a n t

fn a r a t i o n a l way Epion excbasrge tern,

t h e lsst tm vgniehea because $l

-

For

M on the % n i t i d s t a t e .

18 l i k e a a o s t p t i c w m n t t e r n 2n t h a t i t has no pole a s k-

The secand t e w -i.

O; i e is of lcw

angular wmentunr and slow emrgy bebavfor; i t can be t&ea i n t o the bsck;ground, Leaving out all such w m n t t e m , 8s they only e o n t r a u r e t o undefiaed backgremd we find the pion exchange t a r n c m be w r i t t e n i n a gauge invariant f @m:

"pies exefrmge tern"

(for na,ew

0 and &g is raplaeed by

favarisnt q r e s s i a n )

@,gl

.

-

eZ

SO

we ~ t f l l have . a gauge

Zt f s w i d e n t t h a t the ""pan exchange ternft hhaajr no p r e c L e defingtion (sme parr of the nucleon

resonance t a r n is added).

'Ebe &Iguity

of e m c t l y

what p a r t i e a c a d w c because the d i g u i t ; y is fslwly v a q f a g and e a s i l y Elbsorbed i n the deffnftion of '"background". caae, a t any k2 (not only k2

We would suggest f o r a getneral

0 ) and my i n i t i a l & f f n a l s t a t e t h a t we take

j u s t the t e r n

.rf fnli> 5s the m l i t u d e f o r the d i a a r m

Since s t r i c r l y t h i s is off s h e l l s-

d e f h f t i o n by theory m y be used, althawgh

a l l we c m be sure of Is t h a t the recliduie on &elf i a correct, term'qo~es not have

2

The

""no

exchmge

). The mlagular dependence is slw and shows no

forward peak. Any form l i k e

f a r m y a would be af l righe, but (8.5) has the v i r t u e t h a t tbe @Angular bekavior

M

k

+

-+

O s p h y s i c d l g correct,

I have gone t h r m & t h i s long d i s e w s l o n t o shaw t h a t Walker c no tXsaoretical e r r s r of principle i n h i s etbocrl of f i t t i a g .

rrdjuatabls

parmetere of f i t were the desZred quaatitfear t the mass

U The second factor

the photon polarization,

3.8

P

e

a@

and

~ t a ybe

2

of the

written

y ~ ~ d x l ~ a c ivector on af the vector p a r t i c l e .

2.3

(There i s no generally accapted emvention f o r haw t o w r i t e One way is t o write m

F@. 2 m /Zy @

Q,

so y+ is

l

2/g

0

4

-

F+ f o r what we have c a l l e d F+, others w r i t e

gm, but s t i l l others use the same l e t t e r y

Q

f o r another

way of expressing the coupling.) The exper2meaterer avoid a l l t h e ambiguity by noting t h a t the d i a g r m a b w h p f i e s rhat a f r e e

e e . C

(p

=son would have a c e r t a i n r a t a t a disintegrate i n t o

n e y give t h i s r a t a ( h i e h c w s =re d i r e c t l y frora experimnt m p a y ) efe-

+$'l

=

1.36 t .I x 10~'GeY

e i t h e r d i r e c t l y or as a braurching r a t i o , c m e c t i o n (for any vector meoa V of ( a =1!137)

as

For the

P+ = ,080 (GeV)2 o r @:/4

(.

A s b p l e calculation shows the

neglecting electron

-8

n

-

-8)

13.3.

One can look i n t h i s energy region f o r other products (such a s 3n, or rtoy) and find the resonance again

- thus detenngning the hadronie m p l i t a d e s f a r

Q33n o r @noy, w u a l l y given in the fom of branchinp r a t i o s , Tn t h e case the f i n a l e t a t e i s 3n there i s mother reamance a t 782 &V, the

IJ

mson

- it is studied i n tha same way,

AgaZn studygng 2n fknlnaf s t a t e s a l a r g e resonance near 765 of width aromd

125 &V,

This l a r g e width a&ee con@%derable& L p i t y

(i.a aaawptione of how

2

I' v a r i e s with q ) i n d e t e m i n i n g t h e constants, -a

and width,

t r i c , sharper on t h e hlgh errer@ sfde.

A

Further t h e

smll shoulder is

- t h i s e f f e c t i s i n t e r p r e t e d a s i n t e r f e r e n c e with the

nearly apparent

resonance aslsuraing there i s a f k n i t e a w l i t u d e f o r w 2 a .

w

(Thle v i o l a t e s i s o s p h ,

tlnd i s an electromagnetic e f f e c t which we will, d i s ~ ~ 3i n~ the 3 next l e c t u r e , ) Values f u r t h e various constants f o r t h e vector mmon m y be found i n t h e P a r t i c l e Properties Table,

There is now s o m a d d i t i o n a l d a t a from t h e

Oraay Storage Mngs (S, Lrrf rancois , 31971 Z n t e r n a t i m s l s p p o s i u m on e l e c t r o n and photon i n t e r a c t t a n s a t high a e r g y ) ,

Sam of t h e differences from P a r t i c l e

Table r e s u l t s a r e new data but so= a r e due t o an a l t e r e d way af reducing data, especially for the p,

We s h a l l have t o wait f o r new p a r t i c l e tablea t o thrash

these d i f f e r e n c e s o u t ,

B New Q1

P a r t i c l e Tables

f 2 * 1 t .7) x

B ( + + ~ B - )= 46.4 2 2.8%

~(4yrOy)

('25 2 ,091 X

B($-*%$)

B(++Jn)

(14.7 2 2.2) x

B(++3n)

m

~(ur.n'y) = '07 2 '02

ru total. = 4.2

.O4

B(ws2n)

r tube%-> 780

mp2 l"

P

r P-*C?+@-

r

,02

1.0 NeV

2 '02

(Phase 87' t 15')

6

total. = 153

6.0 2 .3 &V

BCWSR)

= 90 2 4%

rw

= 11.4

B(w2n)

.c

~(.-+e'e-)

= ~ 0 6 6t ,0017)~

P

3

1 3 &V

6.5 t .5 keV

= 2.27

2 gw/4n =58.3 2

g( /4n

-

755

nt

l'

13.3

P

+

+ 0,9

,009 2 .Q02

10

t o t a l = 125 2 20

r P+e'e-

C a f e d a t i n g from t h e l a t e s t Oreay d a t a , we g e t 2 gp /4n

= 18.2 2 5%

~ ( w - . n ~ y =) 9.3 "-22

.76 2 .08 k e ~

2

35.4 c 4%

2 , 7 MeV

~ ( @ q O y ) ~ o seen t

p

Branching r a t i o

BC#+n"y)

rg t o t a l = 4,7

w

-

7.5 2 .9 keV

85

Vector Mesons and Vector Meson Dominance Hypothesis but Clrstzy snaking "corrections f o r f i n i t e widthf' "reduces t h e data i n some o t h e r

Nota,

Now t h a t we have d a t a f o r t h e r a t e s

and

W T I ~ we

can compare t h e

predictions of t h e quarlc model with h a r m n i c dynanaics of Femma, U s l i n g e r

and Ramdal. Orsay

Quark Model r @ s y = 1.73 x furny

GeV

I , O 2 . J 4- 1 0 " ~%V, .6 f .2 x 10"~ &V.

1.92 x 10-'GeV

W e will make a digression frm our m b n discussion of vector =sans consider t h e i n t e r e s t i n g f e a t u r e of w

-p

to

interference.

The reason w can go i n t o Zn is t h a t t h e s t a r e / w > is not pure / w a > (U;

4'

+ dhj

isospin zero b u t has a small admixture of i p

Lsospin one i n i t ,

p

- 2a

: U (

- dr)

This mix%ag is due t o e l e c t r o d y n m i c s .

We consider the w

tude t o be i n a

2

-

-

p system a s a two-stare system where Cpo is t h e ampli-

s t a t e and Cwo i s the m p l i t u d e t o be i n an wo s t a t e .

the =ass n a t r i x f s

therefore the t r u e w i s

E16

Photon-WadranInrmetions

6 can be given i n terms of t h e branching r a t i o f o r w2r

Frm Orsay e x p e r g w n t a l d a t a the abow branching r a t i o Jls 4 f 2X we t h e r e f o r e

obtain

14

3 * 7 1 .9 &V.

by fitting t h e "shoulder"

The phase 55 the w-p i n t e r f e r e n c e is detennkaed ia

+

(@'P

m a r the

+. A X-)

I f 6 is negative t h e phase of & / ( l 9

is 87' J: 15'.

p

resonance; t h e value

+ 561)

is l09*, which is

in f a i r agreement with t h e a p e x h e n t .

We can understand t h e rolxing a s due t o e m r g y of

WO

e f f e c t s : ( l ) The electroroagaetic

is not e q w l t a t h e energy of dB because of t h e d i f f e r e n c e i n s e l f

energy of t h e objects and t h e d i f f e r e n c e of energy of %nteiactlon, (22 m e r e i s a c c n t r i b u t i a n d w t o a m i h i l a t i o n p.

+

y

-+

g,,

wo

+

y

+

wo go

+

Y

+

w

.

We can g e t estimates f o r the two above e f f e c t s from t h e knowledge a f K*'

+ p -'p

-

.

Let the s e l f energy o f d be a mass d i f f e r e n c e s and from P P and t h e matual energy aE d x be -b; then the s e l f energy o f u is 4a tund t h e

K*'

and

m t u o l energy of Prom w

=

a 1

U ;

is (m;+

- Sb because the photons a c t twiee on double t h e charge. - (ug - -X], P+ = uZ,K*+ u s and K*' = d z dz), p.

-

K

we obtain t h e f o l l o d a g electrotaawetic s e l f energies

But we a l s o have a raatrjjr e l m e n t b e t m m 1 3a (4a a) 2 (4b b)

- -

from t h i s alone the mass m t r f x i s

-

- %/Z

Vectsr Mesons and Vector Meson L)smin'naneef-lypefhest's Slnce

K*'

-

= 3a

+ 0 p - p we have 3a

- 3b/2

+ 3b

= 9b/2 m

(K*'

- K*') - (pi - p o t )

Note t h a t i n eq. (15 -5) we have w r i t t e n p" f a i n d i c a t e t h a t p''

instead of p',

t h i s is only

does nor contain the contribution from t h e a n n i h i l a t i o n

t e r n which we c a l c u l a t e next,To f i r s t order i n e

Let x be t h e amplitude f o r

2 42'

is x i 3 (see l e c t u r e 16).

(U;

- d3)

+

2 the change i n ~ s s s18 given

, the

y

amplitude f o r

--1

42

(U;

+ dz)

The maa8 aratrix due t a the a n n i h i l a t i o n term is therefore

proportional t a

We already have one -try f o r the full matrix

+

of t h i s matr%x, Am

61

1.53 lvleV we therefore have

Addlng the contributions from the electromagnetic s e l f arass and f r m the annihilhtion t e r n t o the non-diagonal matrix e l e w n t we find

and since

- p'')

-(p'

6

-

--

-l,02 &V

+

f r m data i n the p a r t i c l e

-2.4 .k 2 . 1 &V,

- K*') -

+ (K@

.5 &V

6

(p'

(p

C

- pot)

(U ,fO)

- p') - 1.53 EleV

- K*') - '0( - p)' tables, (K* - K*') -8 +3 (K*

E(eV

and 'p(

-)'p

But we do not t r u s t these r e s u l t e , especially the p mass

difference, we oaly say chat they Pndicate t h a t 6 i s l i k e l y t o be negative,

P e 'e 4 t t u

The yp coupling of the form

'

i s not gauge invariant aod

would give a f i n i t e massZ f o r the photon v i a

The coupling i s A P,

f o r exmple.

However ~ Bwe ean~ couple ~ the~ f i e l d s P

~

u'J

of the photon sad

X t leads t o

which i s evidently gauge invariant.

Since eye q is always zero we aee our

pole behaves l i k e @I

The l a t t e r term has no pole s i n g u l a r i t y and is "lost" fn the background of other than pole t e r n and e f f e c t i v e l y t o eY

e P t i . e , 2 C nr

P P

WC?

can use a restdue just prapartionaf

2

=

Vector mesons (continued) Me n m return t o our discussion of vector m e s w ~ . We can see what %U3 2 given a t the end o f l e c t u r e 14, Wtz gives f o r the r a t f o s of the couplings

89

Vector Mesons and VeerorMeson Dominance Hypothesis use t h e s i w l e quark model ( f o r c o w t i n g only) and assu&ng t h e Q,

18

purely

J couples t o @ p a i r proportional t o i t s char*

strange quarks.

4

88

+

- 113

- 3-13

Noting t h a t these a r e values f o r the coupling F, euld t h a t g is t h e r e c i p r o c a l we f i n d -2

gp

:gu

-2

:gI

-2

9:1:2

Various p o p l e have t r i e d t o c o r r e c t t h i s f o r SUS breaking, but nobody r e a l l y b m s how,

The f i r s t is, t o what e x t e n t Is

There a r e two questions,

The low valae oE the Q1

I s e e no way t o d e t e r d a e t h a t .

t h e ip pure ?:a

brsnchinlng r s t l o is i n t e r p r e t e d Fn the

+ 2%

m d e l by s a y h g t h a t the 4 being

made of purely s t r a n g e quarks f h d s i t hard t o go i n t o non-trange

objects

(S),

I f so, a l l t h e ~ l l t u d at o go t o ny coaea frorn an a a x t u r e of t h e w s t a t e , (U;

the

+ dx) /fli n + comes out

the 4 wave functfon,

in

t o about: .l0, although t h e r e is so= uncertainty 5: .03 a s t o

how =ss f a c t o r s e n t e r .

Prablern.

I n t h i s way the m p l i t u d e t o f i n d

t h e s t a t e is 99X pure ss,

This i a only 1% prob&ility,

b k e a. theory t o e s t i m t e haw much Q,

+

3n would be expected i f

i t i s due t o t h e f a c t t h a t 4 has a small admixture of u s t a t e ; assurnfag (for

no e x c e l l e n t reason) t h a t

8;

c a a o t go t o 3~r.

&re importmt i s the question of SU3 brealring because the m s e e s a r e not equal.

Par one e x e q l e of m b i g u i t i e s , s k o u l d we compare P 's d d i e c t l y with

, or

the $U3 predicted r a t i o s , o r should i t be \/mv should bear the s i a p l e r a t i o s ?

P /m v

V

o r what, which

$U3 cannot t e l l us and nobody r e a l l y laxows how

t o c a l c u l a t e i t , although varioue guesses have Been made.

(I p r e f e r F,/%

f o r a t h e o r e t i c a l rewon, but one which is r e a l l y not profound o r necessary). These questions probably do not strongly a f f e c t the p,w r a t i o , a s t h e m s s e s a r e close together,

As you sec? the r a t i o 9 : 1 5s not very Ear o f f , (and, i n s p i t e

of the u n c e r t a i n t l e a , the cp i s a l s o about right). Vector Heson Dominance Model 4.

To s u m e r h a we e x h i b i t the photon t o n

n

- w l i t u d e near

t h e pole of t h e

p (the only m e of the three

when q2*

4, u,

P

which couples t o the pion) a s

h e would ordinarily expect other terms (such a s a d i r e c t

HL 2 e

P

caupling of photon t o pions or various other ""tntemdiate to this.

statesf') t o be added

Tiris, of cauree, would a o t chaage the behaviour near the pole ((trrtrlch,

unfortunately f o r the p is not so definite a s the width Pp is r a t h e r large, whicb tnakes mny d i f f i c u l t i e s in practice i n f i t t i n g i t , nevertheless i n t h i s thaor e e f c a l discussfan we s h a l l neglect i t ) .

Aslother way t o say i t is t o suppose

2

the n w r a t o r has a f a c t o r F;(q ) which varies with q

2

again t o say the "constants" f p n z o r gp vary with q2 m

P

being defined.

such t h a t &(m

2 P

) = ,l or

only t h e i r value a t g

2

All these ways say the same thing so we s h a l l not argue

about them. Nevertheless a bold hypothesi.8 has been suggested (vector =son dominance) t h a t t h i s expression with f other t e r n ,

(Inn

/g constant i s a l l there i s t o i t p

I see no good physical reason for' t h i s ,

another mystertortkl hopothesfs of the 8an analogous thing with a pion pole a t the pole,

m

P

m q2

-

- there a r e na

It i s made i n analogy t o

kind, BGAC, t h a t works,

- but only extend i t from g2

melee we do

mwZ

m

.Q2

0. Here we suggest eq. (16.2) i s v a l i d nor only near q

2 but a l s o f o r a l l q o r a t l e a s t t o q2

0.

W e b o w f a r long wave leagths the pion looks l i k e a s h p l e point charge

ao t h a t ss

2

-t

C? (use crossing t o cooverc (16 '42 Bp (q -mp 1

--

IN>

IN'

p with p o f a r i z a t i m U.

The term < M / p

IN>,

the coupling o f a p to a nucleon

is defined only a t the p pole, and there has an e l e c t r i c and ntametic part s o

%NI

we c m w r i t e as

+

C~

1.

.

IN)

(y$-dyy)

Thus, expressing the

ksovector

part of' the current coupling i n t e r n OE the txswllly defined form factors w e have

,

i s j u s t J'

m

-

hence f o r e m w l e

Now obviowly w e suppose the three pieces i n t o vhlch

it%

a r e the corresponding vector m s a n resoaant pieces of (17.5).

s p l i t i n (17.6)

This inay be w r i t t e n

i n a very s i q l e way, f f t h e gV r e a l l y have the r a t i o s of the quark m d e l (Lecture

14) we a r e saying, of course

the sm f o r all q w r k s is

where t h e constant,

Bowever

M

.

s t i l l must define i n an e x p e r i m n t a l l y d e f i n i t e way what JY, J ~ Js. ,

W e can use i s o s p i n ,

o r J P , J@,J' are.

hypercharge and quark number (or

) a s three conserved q u a a t i t i e s (each d e f l n i t e f a r a

r a t h e r , baryon n-er@ particular transition X

+

Y) t o serve inntead of quark nmbera. P

represent t h e current of Z-component isospin;

J

m -JZ

+$

JY

to write

+

- JY + ,IB

Js

quark, IZ d 113 f o r t h e currftnril you g e t one f o r JU, zero f o r J and

(beesiluae if you erubstitute f o r e x q l e the guaaturn n d e r 6 of t h e

+ 112, Y

-+

11'3, B

J", e t c , )

Hence J"

'

cvrrent of hypercherge, J~ c a r r e a t

of beirym n u a e r (equals l 1 3 tims t h e current of quark n*er)

Jd

Thue l e t J

...FZ

m

U

lPlis then deflnes precisely what we m a n by t h e currents h (14.5) and (14.6). f

l e , l e t us apply t h i s t o the K K4"

couplinge of J' t o K

+ 1/2, Y

-

C

1, B

(we a l s o calculate .lP,

-

decay of the 4 .

.JW

We need the

f o r cnnpleteoess).

For the K+,

O, sa a t zero q2;

m u s e o & i a b g t h i s with ( 1 4 . 9 ) and (14.7) we see we a m t have

defining how the t o t a l of ( l ) i n (17.41 is partitgonad,

2

which agrees p r f e c t L y with the f4 m/4n

-

1.47 from

(

A t my r a t e we predict

4"

-+

K K- and g

4'

2

/4r

13.3

f r o m 4 + e\-.

How slow is 4

-+

3n cmparczd t o w

-+

3nl

(p

goes 18% Co 3n, but I' i e only 4

so the p a r t i a l width i s only .7 MeV while i t is B? &V is l a r g e r f o r the QI but -re t h e ~ ~ l a t r fe xl e m n t varies,

for the

W.

Phase apace

detalXeZt ana1ysSs would require knowledge of how I f e is the polarization of the vector nreaon

llnd PI, p2, p 3 the four vectors of the three pions the emplitude must be constant eliYap eli plv p2@ pgp.

The "constmt" f o r the 4 has ta be about 0.1 a f its value

f o r t h e w.

Lecture l 8

Ismss The 41 appears ea act ast pure

L.

&at is the significarxce of a l e

- or

r a t h e r how can we d e f i ~ e& i s h t e r m a f q w t m nu&trra o r r u l e s without

9.5

Ymmr Mesons and Vector Meson Dominance Hypothesis r e f e r r i n g t o the quark m d e l ? We don't k n w how,

Wcr t r y t o say t h a t i t w u p l e s

weakly t o s t a t e s which have no strange p a r t i c l e s i n them, s o not t o 3n but yes to

E,

Yet such an idea "a s t a t e t h a t has no strange p a r t i c l e s i n it" $8 not

readily definable.

has no o v e r a l l

Due t o s t r o n g i n t e r a c t i m s a s t a t e l i k e

qwntuat a m b e r s t o d i s t i n g d s h i t from 3n; I n f a c t v i a v l x t w l i n t e r a c t i o n s

-

should couple strongly t o 35 ( f o r example through a tairtual w, KK

the

.g-*

u

3n.I

m e r e f o r e the question r e m i n s , what keeps the 4 Exono coupling strongly with 3n'l From the paaitat of view of

i s s e l e c t e d from the

~1

quark =del we &&t t r y t o say t h a t the

l -&ZM~] state

state

of the w simply by the f a c t t h a t s quark@

carry a l a r g e r =ss (or a d i f f e r e n t Faceraction energy) than non-strmge quarks, So i n lowest order of perturbation theory t h e e i g e n s t a t e s of energy a r e and

1( u ; ~ z ) . G

But then we f i n d these s t a t e s a r e unstable, and a s a kind of

perturbation they decayiy,i t turns out wItb s m l l widths.

- there is a u coupling equal t o - L is n o t small

s t a t e s f o r e x m p l e Qt

coupling of 4' and o0 t o every meson s t a t e (with t h e of t h e Q, coupling).

fi-

+.@E

But t h i s ""perturbatione1

-*

w

Thus f o r these v i r t u a l mson

there a r e d i a g r a w

4-

W

l? which mix t h e 4, w s t a t e ,

These d t a g r w cannot be calculated, (the above

diverges quadratfcally) and no w l c u l a t i m of such v i r t u a l strong f n r e r a c t i s n s h any problem has ever been successful.

But usually the q u m t i t a t i v e idea

t h a t i f the s t a t e s can be connected v i a strongly i n t e r a c t f n g virtuin1 s t a t e s they can go i n t o each other is v a l i d .

The coupling constants £+E aad fMa a r e s o

large they would seem t o have e f f e c t s of order I, t h a t is, t o strongly m i x the o r i g i n a l pure &a and

1(u%d;i).

a

Of course i t Is always possible t h a t t h e world t h a t the cmbination

as is not

18

s i w l y complicated,

i s o l a t e d by p r h c i p l e i n the beginning; but the

vex)r v i r t u a l i n t e r a c t i o n s s e l e c t s o m l i n e a r ta&ination of be t h e

I$

[and the orthogonal c*ination

the co&ination Is

and

1 (UCM~]t o

&I

f o r the w) and i t j u s t coraes out t b a t

L,

Then perhaps it %a r e a l l y possible t o tmderstmd why 4 but is the a~lcllbtlessof 4

-+

.c

ny i s eo sraalf,

3%r e a l l y obvious I n t h i s model of the s t a t e ?

But s r e l l @ore s t r i b f n g is t h e f a c t t h a t t h i s "accident" %a repeated again!

+

For the mesons of spin 2 where the Z(l25Q) goes nicely i n t o 2n or 4r end i n t o

S. The f(1514) goes p r e d s d n a t e l y i n t o '&l

and even h t o G%-k*and not i n t o

nn i n s p i t e of the even l a r g e r phase apace.

(The quark w d e f r a t e s (Fep-n,

-

U s l i n g e r , b m b l ) using pure l ( u s d x ] f o r the f ( 1260) and a; 42' a l l high, but the relatkvlve proportions a r e right.) m e e x p e r i m n t a l and quark m d e l

NOTE:

(m)r a t e s

f o r f (1514) a r e

+

f a r 2 w s o n s are;

There are i n addition two hadronic phenomaa ilesociated w%rh the 4 unless the QI Is, l i k e

vhlch axe =expected

+ 88

+ nn

N

thou& the 4 were not coupled t o the M, N* (a aucleon

trajectory) j m c t i o n , (relathe to p

+N *N ++

The f i r s t is badward praduction of rfr i n r B N

of zero s t r m g e a e s s ,

It: %B very weak,

weakly cowled t o hadronic s t a t e s

,S ;

AgaZn p 4-

p + Q1 + a n

is very etrangly suppressed

for exmple) a s thou& the $B%

8

a d

carnot be lazade fro=

the non-strange quark@ in the a t b e r p a r t i c l e s ,

VDM aad Photon XIadron Interactions b n p l e t e vector diminan= i ~ l f e tsh a t the photon carnot i n t e r a c t with a kadron except a s i t f i r s t becoms

et

p , w o r $B,

i n general) i n t e r a c t s with the hadrons,

and then t h i s

p, w

o r 4 (say V

Thus we would expect t o f&ad so= kind

of r e l a t i o n beween awXitudes l i k e h p (7

f

A

+

B) = v

%v

@ + A +

B).

There a r e several, p e e t i w involved i n using the f o m l r t . are t h e o r e t i c a l questioas of - w a g .

Since y has

F i r s t , there

= 0, the vector meon

97

f/eeforMesons and Vector Meson Dominance Hypothegig 0, f o r which s t r i c t l y

an the right s i d e should be a v i r t u a l vector meson of q2

It might be argued t h a t sixlee -ss

no d e f i n i t i o n can be given,

s t a t e inco&ag p a r t i c l e miglrt have the least e f f e c t a t high

on the i n i t i a l (here E

8,

L=:

p

a n T a y f o r f i n i t e glass) t h i s m l a t i o n m y be mst nearly correct Ear r e a l vector -sons

on the right-hand s i d e a t very high

S,

b o t h e r related d l g u i t g i s

t h a t the photon has only WO golarizationa, h e l i c i t y

f;

l, so the equatloa has

But h e f i c f t y i s

only m a a i n g f o r the corresponding h e l i c i t y ?: l s t a t e s of V ,

not a r e l a t i v i s t t c a l l y invariant concept and depends on the Erme.

Most

t h e o r i s t s have corn t o the conclusion i t is i n the s-chmnel f r m e t h a t the

is reduced by going t o

p h e l i c l t y should be 2 1, A t any r a t e t h i s -certainty

Thus we s h a l l , t o avold auch c o q l i c a t e d discussion, limit our-

large s also.

selves t o c a p a r t s o n s t o experiment a t the h i @ e s t energies now avallsble. The next question is how we a r e t o o b t d a the vector =son or mplltudes V

+A

-P

crass sections

B, a f t e r a l l V mson beerne a r e aot available,

Sowtims

t h e o r e t i c a l a r g w n t s are available, but i f they a r e too complicated they a r e The m a t useful s i q l e cases a r e :

not uaeful t o t e s t WM.

a) Pseudoscalar =son production, i n p a r t i c u l a r p b) Biffraetion ( e l m t l c ) s c a t t e r h g from nuclems p

+ Nuclaw

J.

+N p + N

J.

n

+ N.

+

p

+N

o r nucleus

+ Nueleus,

p

kle discuss each i n turn, We can study the r e a c t l m p

reversed reaction n

+N

+

has been studied a t L5 &V

p

+ H,

+M

nap

+. 'p

n

+ 1\1

by experimntally s t u d y h g the

&ich should have the @ m e-2itudia.

a d reported by D.W.G.S.

This

Leith, fieno-oloa

Naturally the p is not observed d i r e c t l y

a n f e r e n c e , l971 (Caltech) p. 555. but

+

C

n is inferred fro= s coarplete study of n- p -, n v- a loolting tn

the appropriate Hltlse region of the two outgofng piona. be made f a r palra of pions, say i n mutwl v i r t m f p decay.

However the data a t low

S

Sow corrections mmt

waves, &icb a r e not due t o t hias been nlcely malyzed i n d e t a i l

t h e o r e t i c a l l y and we caa describe the r e s u l t a , and empare t o W# e x w c t a t i o n s , We evidently wish t o c m a r e these r e s u l t s t o reactions lib y

and

y

N

+

n N i e r e l a t e d d i r e c t l y by W t o nip

acleaaxl l i n e a r co&ination of P , we drop i t , and the w

M,

+*

interference.

-*

V'

n where V'

-+

W+

is a vector

Sinctl the $ coupling to =sons

mM

+p

is smir

then predicts f o r e x q l a that (by

+

a r e "fnsity r a a t r b e l e w a t s t o proJect out the h e l i e i t y 2 f reactions. -2 -2 is about: 9/1 so the second t e r n is Data i n the o reaction a r e lacking, but g P /gw (pll]

probably never large. 10 times

do

be a s -11

p

((n-

+

I n addition, a t 8 GeY

-r

p' n) is i t s e l f about

w n) i n the f a w a r d direction, so the w contribution might

a s 1%t o the

p-w interference

(E' p

mM

result,

The s m could not have been s a i d of a

term. ( f o r an a w l i t u d e of order .l i n t e r f e r i n g with 1 can w k e

a 20% e f f e c t , b u t i t @square is only B ) . m a t i s the reason t h a t the c q a r i s o n i s made t o the suzn of yg

+

+

n. a and yn

+

n-p croas sections

- for i n that a m

the p-w interfereace t e r n (from isospin considerations) cancels out ( s s doaa the P-+

interference), The cross sections f o r

4 3 1 1

the h e l i c i t y c a b i n a t i o n s of the p have been

aaasured (and reported %XI the f o m of density matrices) bp measuring the angular

+8-

dietributiona of the p a i r of pions n the polarized p

2

.

r e s u l t i n g f r o a the dlafntegration of

We a l s o have d a t a with polarized phot~r-e(perpendicular and

p a r a l l e l t o the plane of production a d s o can make two camparisona.

The vector

dminanee model predie ts then,

where daldt (yN

+ nN)

m m the average of dafdt

(m

-,-P) and dcldt (yp

and the p % are densktg nnattrix e r e m a t s i n the h e l l c i t y f r a ,

3.

.+

E

The r e e u l t s of

such a corrtparisan appear aa, follows (polarization data f o r photons a t 15 &V a r e not y e t available, but a p a s s e d extrapalation front 8 &V was nzade does not a f f e c t the ccrlnparlson of the s m of

5

8nd

3,

mmured as the .ctnpolarlzed cross section a t 15 GeV),

- this

which is of course

n)

98

Vector Mesons and Tdecror M e m L)amr"naneeWvmthesis

Kt &Vfc

c t

The r e s u l t s axe seen t o agree i n eexaety the f c m a r d d i r e c t i o n and generally f o r the uonatural p a r i t y exchange oil

.

, but

c l e a r l y disagree away from t

-

O for

Thus YOH a s s general theorem cannot be exactly c o r r e c t (barring the possi-

bZlity a f an mamw w c o n t r i b u t i m i n t h i s region], %W

photoas do not couple exactly ae off =ss

J w t how do they couple?

To make progress we should study t h e nature of the

deviations front a s t r i c t VDH and f i n d t h e i r c h a r a c t e r i a t i c a , we a good theory, which a t present we da not have develop

- we

But

sM.11 vector =sons.

For exmple! h&

- but should be able

cottld say why these various c r a s s s e c t l agree ~ ~ ~ for

to and f o r

the f o m a r d d i r e c t i o n , and why not Ear %? Perhaps s o m clues can be obtained from t h i s exmple s o we study i t f u r t h e r . The values of t we a r e studying here a r e sa small t h a t s i n g l e pion exchange ought t o d o d n a t e both cross sectionrr (Lee.

y

a s well ars p).

T f tbie

were e m c t l y t r u e without a b s o q t i o n corrections we would expect a r e l a t i o n between the cross sectiona s i m i l a r t o (18,2), (18,3), o r (18.4) simply because

The only difference (aside from t h e massZ dlffe'rence of o, y and selection

aE t h e c o r r e c t h e f i c i t y m p l l t u d e s t o c m a r e ) would be t h a t the y is coupled r e t h e plv

+ pZy of

t h e pions v i a a f a c t o

we would expect a r e l a t i o n l i k e

and the p v i a

Hence

t o replace (18,1), and corresponding r e l a t i o n s f o r the cross sections ( l 0 6 % ~(l8,3), (18.41 with g

P

replaced by fpww2). m e r e f o r e insofar as these r e l a t i o n s agree

f o r small t we have not made m y new r e s t of the vector dominance r e l a t i o n s other than t h a t fpwT2S gp2, a t e s t t h a t we have already seen works very well. t h a t equality of c a u p l h g constants w a s

cul

If

accident then the law t agreement

k.:uid only m a n chat s i n g l e pion exchange dorafnates both reactions a t small t

-

somthing we expect t o be t r u e on other grounds. On the other hand even a t low t , fi

deviate

- how could t h a t

m% f o r example, the two theories

corn about? The one pion exchange term aim t , of course,

b e corrected f o r absorption,

The differences between o

yH

and a

at low

&

met

l i k e l y c m be attributed t o differences i n the degree t o which a b s a r p t i m uludifies the m e pion exchange ezpectation. In the f i r a t place (see LE?ith% report) a f a i r f i t t o the p d a t a is given by a one pion exchange model corrected f a r absorption (due t o P.K.

Phy;s. Rev,

W i l l i w ,

1812 (1970)) f o r the a w l i t u d e e t h a t concern us t h i s gkvee

and

We have already sees t h a t the anabsorbed one pion exchange gfves f o r theere m p l i t u d e s { i n connection with the discussion of photaproduction)

-

This leads

and t h a t i d e a l absorption would subtract 1112 from each m p l i t u d e , t o zero A,

i f -t

:m

and f o r large t equal amplitudes f o r

Al.

All

and hence

t o ail asymmetry t h a t r i s e s from O a t t = O t o l a t t = mwZ and f a l l s beyond t h a t t o 0. photon,

% i s agrees with the a s

t r y observed f o r the Q, but not f o r the

1%the photon case the f a l l off f o r large t i s only aodazrate, t o say

0.5 o r so,

Thus f o r the photon i t looks l i k e a more appropriate correction

f o r absorption lnfght be t o subtract -re

l i k e 213 from the two mplltCLdes.

t r y i s 1 f o r t = 2mnL and f a l l s t o 0.6 f o r large t

- not

in

Vector Mesons and Vector Meforr D O I P Z ~N~yZp ~ f~l ~t t C ~ iCs We r e a l l y do not exactly k n w why the sub-

disagreement with experiment).

Possibly i n

t r a c t e d t e r n f o r absorption should always be the f u l l 1 / 2 ,

t h i s case e x t r a contributions (isoecalar o r isovector?) a r e contributing t o l%e point

t h e photon case t o increase the 112 appropriate t o t h e p case,

here is an ef f e e t d i f f e r e n t f o r p and y and hence df f f e t e n t Era= expectations of VD& (unless i t could possibly a r i s e from the u term).

Emever the c o r r e c t

c o n ~ t a n ttern t o use i a detemined e x p e r k n t a l l y by the cross s e c t i o n f o r t

3

O which is j u s t proportional t o t h e square of t h i s constant with known

coefficients,

This dete-nes

But a much =re

t h a t the constaat is indeed 212 t o 16%.

s t r i k i n g difference is the difference a f a

This difference is already l a r g e a t sml1 t

- the oB f

and a

YZ

EL'

drops o f f much Easter

Why? X dont t know. X thlnk the onystery here is $ay the cr YL* P1 f a l l s s o f a s t , In t h e E i t eo OPE plus a b s o r p t i m i t c~asfound empirically than does a

2

that a f a c t o r e"('lraz

needed t o be included i n these cross sections.

P. K, k f i l l i a m had suggested such a f a c t o r f o r absorption e f f e c t s but expected a much s l n r e r f a l l o f f ( l i k e e3('em'Z').

To what i e sueh a rapid f a l l off duel

Such a rapfd v a r i a t i o n is e n t i r e l y unexpected, method of analyzing t h e d a t a , m's i n a m u t w l s-tprave (not a

near

-t

-

It is very possibly due t o the

The e f f e c t of an m p l i t u d e t o produce a p a i r of 1))

has t o be subtracted away,

The value of

mzZ is e s p e c i a l l y s e n s i t i v e t o what is done here ( a t t

-

p

It

O o r -t

2 10mn

i t Is not s e n s i t i v e , ) Nevertheless there is a contribution t o n a t u r a l p a r i t y

exchange which is d i f f e r e n t f a r y and p ( v i o l a t i n g VD&). theofy of &ere and when such deviatgons should a r i s e ? probles t o analgze t h i s i n m r e d e t a i l .

How can we m&e a @e leave i t aa a

Onet obv%ous p o s s i b i l i t y seeing the

l a r g e n d e r i n the elrponent is t o be reminded t h a t t o t a l absorption e f f e c t s

(as seen i n e l a s t i c s c a t t e r i n g ) do f a l l off a s expbt with b of order 8 o r 9 f o r n nuclean s c a t t e r i n g .

So perhaps a c a r e f u l anaZysis of absarption, o r

a l s o of t h e p o s s i b i l i t y t h a t the source BE t h e pion is i n d e f i n i t e by the s i z e of the nucleon or the p, w i l l explain it.

The point of explaining i t , however,

i r s t h a t t h e drop off is much slower f o r the photon c a m .

Ttzerefore whatever

the c a m e i t works d i f f e r e n t l y i n p and y pseudoscalar production,

Zn i t s study

1Zes t h e p o s s i b i l i t y o f understanding physically h e r e t h e ideas of VDH go wrong.

The problem cannot be d i f f i c u l t

- the e f f e c t shows up

f o r smll t and

hence f o r l a r g e i m a c t p a r a m t e r and therefore i n a realm *are

physical

phenomna a r e usually underetand&le,

The next topic we take up is the dgffractive production of vector mesons by photo=.

here, to

A t f g r s t we study the p =son

f o r amre detailed data is available

I w i l l not go i n t o a s m c h d e t a i l i n the r e s u l t s a s we a r e aceudlit~rned

- Ear a

f u l l r e w t report see:

@off: 1971 Zntennatima1 Conference m Electron and Photon Interactions a t High Energy, Cornell, Zthaca, N.P

., (1971).

Our aain concern ~513.be a cofnparison t o V W , The go production i n yN

+

the cross section approaches

B

looks very tauch Like d i f f r a c t i o n s c a t t e r i n g , constant a t hi& energy sad hat^ the typical

dawndencl; of such ~ c a t t e r b g , But how does the photon a f f r a c t Snto a

Qne

m w e r ie provided by V M , from the expected w l i t u d e relation (18.1) we expect 1

(because of. I spin change i t is expected that

@W '

rapidly with Lznergy s o only the p t e r n remains).

+

p%

or

w i l l fall

$ON +

Wat do not know d i r e c t l y the

cxoss section of p' on nucfeoas but we c m e q e c t i t t o be a typical e l a s t i c

.

scatter&ng

G crude use of the q w r k -del

supposing the quarks t o s c a t t e r independently

of t h e i r spin directian m d t o be s l & l a r l y d i s t r i b u t e d inside the

and no

p'

suggests a ( p 0 ~ )p. 6 ( n 0 ~ ) . Thie l a t t e r is not Ixnown by d i r e c t experitoenr but [o(n-#

isospin gives i t as

t o t a l cross section f o r yN behaves,

+

+

W-&)

+ o(%'~

+

aCN)].

It does turn our t h a t the

p ' ~ varies with enerm i n j u s t the way

$ [6(n'~)+

o(n%)

Xn f a c t the t o t a l cross ~ e c t i o ais given correctly by t h i s r u l e and

equation (19.1) with g

Et

-

2.8.

There is s 15%m c e r t a i n t y because the observed process is yN

+

N

C

nf

+

n-

and there a r e uneertafatia;is i n i n t e r p r e t a t i o n due t o so-called bck-type diagram i n a i c h the y becomes two pions Rat a t the p reeonmce and one pion s c a t t e r s l i k e

l

k t o r Mesons and Vector Meson Dominance Hypothesis

i n s t e a d of

A s t r i c t adherence t o VDN would, a s f a r a s I can s e a expect t h e analogue

f o r p s c a t t e r i n g t o 2n v i a

but t h e point is i n estimating just t h e e f f e c t of t h e s h p l e p e l a s t i c s c a t t e r i n g s o we can use our no s c a t t e r i n g malogy, This idea t h a t

yXrl -a p N

tells

W

about pP1

-+

pN pernalts us t o

S

The polarjlzation af t h e p coming out can be w a s u r e d .

behavior i n another way,

I t i s very nearly purely transverse and polarized a s the incident y ray.

This

3s excel l e n t evidence t h a t t h e p r a w s s of e l a s t i c s c a t t e r i n g ( a l s o c a l l e d pomeron exchange) does nor change t h e h e l i c f t y i n the c e n t e r of =ss syatemn. , Were t h e d i f f r a c t i o n t e r n is svlaller s o o t h e r processes axe a l s o

e f f e c t i v e , i n p a r t i c u l a r one pion exchange a t t h e lower energies.

The n a t u r a l

p a r i t y exchange has besides t h e gomeran exchange a l s o possible A2 exchange ao t h e purely d i f f r a c t i v e p a r t ha8 n o t been separated out yet f o r a c l e a r t e s t . , The e x p e r i m n t s show some L n c a s i s t e n c y but d a t a lis a v a i l a b l e .

One wily t o t e s t t h e r e l a t i o n s of VDM i a t o conrparr? t h e forward d i f f e r e n t i a l cross s e c t i o n f a r yN W

-+

W,

%h'is

-c

VX wfth t h e f o m a r d d i f f e r e n t i a l c r o s s sectjton f o r

l a t t e r c m probably be estimated f a i r l y w e l l froin t h e o p t i c a l

theorcorn using quark model estimates f o r t h e t o t a l cross s e c t i o n s f o r W, (The quark model estirnate f o r t h e

aT(@p)

=

+

This gives a f i t with g

P

aT(~-p) 2

p, w

and 4 are:

- oT(w p) f

-

/ 4 n = 2.6 2 .3, 8;

l 5 mb

-

at 5

%v.)

24 t 5 and g

t8

m

22 2 6 .

Only

104

Photon-Hadron Interactions

the :g

seems too big.

It could mean the quark estimate of

-

is too b i g

l 0 lab would f i t b e t t e r .

k c t u r e 20 Diffraction production of v e c t o r m a m a can a l s o be seen, and m r e copiously and c l e a r l y fro= y on n u c l e i , A.

(Referwee: K. C o t t f r i e d "'Nuclear

Photoprocesses irnd vector doninance", Ceornell Gmference, 1971,) d i f f i c u l t i e s f o r the

since the f i n a l 3n s t a t e

%ere a r e

is hard t o m a s u r e , and t h e g

i s s o wide t h a t t h e o r e t i c a l q w 8 t i o n s of i n t e ~ r e t i a e d: a t a sire i n v o l w d .

The

c l e a r e s t case e x p e r % w n t a l l y f e 4 production, h w e v e r much more d a t a e x i s t s f o r the p ,

In these e x p e r i m n t s an p production observations a r e m d e of 2%

production by photons on n u c l e i , of course. t h e 2n is seen a t t h e p =ss, e f f e c t with w

-+

The 4 d a t a

X t is a s

A b e a u t i f u l resmance i n wss of

t r i c a l shouing t h e i n t e r f erence

Zn just a s i n e%- production, it3

now good,

The dependence an h, t h e mass nmber of the

nucleus, gives us some idea of a d e t e r a n a t i o n of g+.

W'

and the absolute cross s e c t i o n p e w t s a

Wowever an tmcc?;rta&nty a r i s e s f o r the r e s u l t s depend

s e n s i t i v e l y on the choice of m w k m quantity, t h e r e a l p a r t of the forvard @W s c a t t e r i n g amplitude f+4. Let aON= xrnfO4/~e is i n s u f f i c i e n t t o determine a l l three q u a n t i t i e s g

2

4

Then the d a t a if

/4n, oO, and o 4s'

g & / 4 n i s its Orsay vaiue 13.3 then we caa only conclude t h a t a m y be @ i n the range 8 t o l 4 aib ( a t about 7 &V) perhaps f r o a .3 t o -,S and o

-

,

possibly a b i t lower tlsan the quark m d e i r u l e s would suglgest, but l i t t l e c m be s a i d . TZze p d a t a i s m r e e x t e m i v e .

Were the l a r g e width causes s o w confusion

as t a what p a r t of t h e data i s t o be a t t r i b u t e d t o p production.

Again we

2

have three parameters g / 4 n , opMand apN. h e g e t s agreement with expectations P i f we ehooee apN 4 . 2 4 (but i t i s n o t well determined). The d e t a gives

-

t h e "good valuesff ( a t 8.8 GeV) gp2/4n

-

2.6 and o

well d e t e M n e d and r e s u l t s depend on i t , )

PN

Therefore the phenomnon of vector

meson productim e x i s t s and behaves closely l f k e a d i f f r a e t i o ~process, and could we11 be i n q u a n t i t a t i v e agreement. with the

VX)PII,

Does t h i s t e s t the

105

Yecror Mesons and Vector Mesorz Dominance Hypothesis Z think not because f o m l a (19 , l ) applied t o d i f f r a e t f o n scatterlstg

mdel?

- nmely

c m be derilred ( a t very h i & energy) f r m ano&er hypothesis

that

c l a s t i c s c a t t e r i n g of the p is much l a r g e r than d i f f r a c t i v e d i s s o c i a t i o n

the

s c a t t e r i n g of the p ( t h e process by which pEii

+

p*N where p* is s o w o t h e r

s t a t e of the s a m isospin a s the p , m a n d .tJhich survives a s s d i f f r a c t i o n diasocfatfon ( a s H% perhaps an*

-+.

o r nM

M%

-+.

-c

W).

Such

r*N) a r e i n other reactions

30% of t h e e l a s t i c scatrering fro@ t h e nucleon s o there i s

no reason n o t t o ass-

i t f o r the p,

Furthemore f o r the p we s h a l l only

have t o a s s m e t h a t the p a r t of d i f f r a c t i o n dissociation t o those p * having s p i n 1- ( l i k e t h e photon) is smll empared t o t h e e l a s t l e ,

a d i r e c t expergmental confimtialon o f t h e f a c t t h a t

-c

There is a l s o

@*H

where p * i s

any iaospin one, 1- s t a t e , produced Iln a d i f f r a c t i o n d i s s o c i a t i v e way does not f a l l wtth energy, For productian on n u c l e i our a s s u q t i o n (of the d o ~ i n a n c eof e l a s t i c

- f o r t h e e2aprtfe comes the a t i r e n ~ c 1 e w- whereas t h e

d i f f r a c t i o n diseociatialon) is ever more t r u e a s A r i s e s from d i f f r a c t i o n from the shadow of

p a r t i c l e s produced by d i s s o c i a t i o n can e o m only fro= its edge. The point

is L t i s riot doainimce of the y t o p t h a t we r e q u i r e f o r

the v a l i d i t y of equation (?19,1), i t w u l d a l s o follow from do&nmce of P

-+

i n the products o f the d i f f r a c t i o n of p

P

0s

nucleon and nucleus.

We can s e e how t h i s works most c l e a r l y i a the case of s c a t t e r i n g from a l a r g e nucleus where our a s e u w t l o n is most nearly v a l i d , considering yA t o P&

-+

PA,

-t.

pft

analyze t h e reverse reaction

gA. -, yA

Instead of

and compare i t

m e high energy pA s c a t t e r i n g appears, i n say t h e

C.=,

a s followap;

The waves of. the Incomfng p c o w I n from t h e l e f t and f a l l upon the nucleus where they a r e absorbed. wave lengths i f

S -,

A s h o r t distance ( a i c h c m be m n y

beyond t h e t a r g e t eay at the datted l i n e the wave

function is nearly the o r i g h a 2 , p wave f m r z t i m with aihole a@ a ffunctioa of

106

Photon-Hadron Interactions

b, the irapace p a r m t e r .

byond t h a t the waves Qowly d i f f r a c t i n t o the

s h a h i n a way d e t e m h e d by Euygens principle.

- alb) is the -1i-

Zf l

tude t o E b d a p a t 6 (so a is near 1 f o r b a m l l e r than the nuclear radius, and near O beyond) the i r p l i t u d e f o r an outgoing wave of transverse m m n t m

Q is j(l-a(b))

eiQob dZb f(q). ikr f(l-a(bl> d%

%

The amplitvde a t P is appronimately

Of course the l gives only the fomard be= m d doesntt i n t e r e s t us

is j u s t a s i f the

p c-

out only across the face of the nucleus.

- it

To be sure,

i f the p ware not a p o h t p a r t f c l e but say rnade of p a r t s (indicated by the braided l i n e ) we could not precisely define everything h t e m of aa m p l i t u d e But

t o give rz. p , but would seed a function of a l l the p a r t s as variables.

c l e a r l y f o r b outside the nucleus the p a r t s a r e i n the aaae r e l a t i v e a ~ l i t u d e as i n a p

- only near

of the p a r t s

the edge is there sow d i s t o r t i o n of the r e l a t i v e behavior

- and 80 a projection possibLble i n t o eoae other than the p.

the a w l i t u d e a(b1 is c l e a r l y t h a t f o r a

p

But

except near the edge; so t h a t elastrlc

s c a t t e r i n g is much l a r g e r (for nuclei, a t l e a s t ) than d i f f r a c t i o n dissociation, 3.

-

How we have noted from experlrnent by e e t h a t tJhenever we have a p present there is an amplitude t h a t t h i s p w i l l d i s i n t e g r a t e t o e+e'

(via a

photon) and therefore there i s a source of the e l e c t r o m g n e t i c f i e l d given by jij

Y xamp. t o find p polarized i n direction P, (Pp P

m

- a current - Hp2/gp).

-

Tlrw In our problem a t each point P there is a c u r r a t source of electro-

- thus the t o t a l q l i t u d e f o r finding a photon

m g n a t i c f i e l d of s i z e $(P) g o h g out with wave vector

gout is

!@(P)

e

-ik@at *P 3 d P,

expression f o r $(P) In t h i s the convolution of e-ik r/ -Zkout *I3 gives r:

1 m ,( B w k2 + mP2 and kautZ

If we s u b s t i t u t e our -ikout"P and e just 0) so we eventually get the

k -k

outgoing a ~ l p l i t d ef8Vtpbton is

where QOut is the transverse p a r t of Lout.

The i n t e g r a l i s the same as f o r the

p e l m t i c s c a t t e r h g so the a w l i t d e s a r e proportional

2

mwt bear the r a t i o bnr? /g

- the @roessections

2 D

Oae can of course carry the w t h e m t i c s out i n more d e t a i l separating

107

Vector Mesons and FlecforMeson Dominance Hypolhesis the z and transverse molenew c a r e f u l l y t o s e e j w t how t h e a s s m p t i o n of high energy i s v a l i d .

It is e m y t o urlderstmd i f t h e d i f f r a c t i o n of the

wave i n t o the shadow i e omitted.

ir; 2

shadow is j u s t e

-

kPr

E

Then the w v e going f o m a r d behind the

(l-a(b)) h e r e the incident p i s of enertgy E a d The amplitude t o ma*e a photon of energl E, transverse

-m ' / 2 ~ .

P

momentun Q, hence longitudinal moment= :k

A

-

E

- Q2/ZE a s

1/2E f a c t o r has been included t o take aceaunt of the r e l a t f v t s t t e

normaltrations f o r the incoming and outgoing p a r t i c l e s .

The m P

d i f f r a c t g m i n t o t h e s h a d w maLes i t m

P

'

'-

1/m 2

Q is wrong

(ae though the

fox p's s d a g i n t h i s direetZon).

It is evident t h a t the poferization of the y L s the s a w f o r t h i s process.

~zs t h a t

of the p

This f e a t u r e has been h e c k e d e x p e r i m n t a l l y very c a r e f u l l y

and is very eytribing, the p p o l a r i z a t i o n produmd by paftlrized photons is nearly purely t h a t of t h e photon. Often i t has been auggesred t h a t VDM p r e d i c t s i t s f o m l a (19.1) v i a a diagrlun Like

V

1

suggehlting the photon b e c m s a v i r t u a l p o; zero q L , and hence t h a t the coupling FD (or

q2

m

P

'

- mP2lg P )

should be t h a t appropriate f o r q2

-

O i n s t e a d of

+

where i t i s measured i n e em. It i s a r e s t of YDH t o f i n d t h a t t h i s

constarnt is unchmged i n t h i e range.

But we*have seen, by consfderation of

the reverse reaction, neglaetgng diffraction dissoeiiation t h a t the appropriate conetant should be t h a t of ra and no extrapolation i s involved. P J. a n d u l a haa3 c o n f i m d my a r g m n t by an. a r g a m t front dispersion theary m d the reduction f o m u l a ,

The a s s w t i o n of neglect of d i f f r a c t i o n d i s s o c i a t i o n e w e s i n h e r e ,

Near

the edge, t h e p a r t s of t h e p may n o t be i n p r e c i s e l y t h e sarne r e l a t i v e Htotion t o produce a p , but they m i & t

s t i l l produce a photon,

It m a n s an i n t e r f e r i n g term of o r d e r ~@*/rn~:

*ere

(Pp* i e t h e coupling of

p*

tlmes t h e amplitude t o make p*

t o photons) su-d

i t mwt be smll co&sg only near the edge.

Clearly t h i s is small.

on p*.

For n u c l e i a t l e a s t

The point is n o t s o much t h a t F

P"

i s sm11 (&%eh i t m y be, a s s o l i t t l e p* s e e m t o be produced by photons on

n u c l e i (vhlch measures

bur t h a t t h e nvmber o f p* produced by p v i a

d i f f r a c t i o n d i s s o c i a t i o n is smll. Other Tests of VIM I f VDH were c o r r e c t we expect t h e photons t o have amplitudes l/% t o be

various vector =sons

s o t h a t the t o t a l c r o s s s e c t i o n o f y t e on p i s l/%&

times t h e t o t a l c r a s s s e c t i o n of each of the vector meons.

(neglecting passibjle i n t e r f e r e n c e e f f e c t s of u and g). t h e p (becauee of t h e 9:1:2

-

o,(pp)

r a t i o of

%'

mis

and o(+p) is small).

We expect

is dominated by

We estimate

ot(@p) by t h e quark m d e l ss equal ~ ~ ( n ' p )fa scheme which we checked

i n our a(yp

-F.

Vpb consicierarians Ixbove).

kle f i n d e x p e r i m n t a l l y t h a t atot(ypb

is too l a r g e by 4QX, a s though o t h e r processes = r e available t a the photon t o i n t e r a c t with t h e proton besides go1ng throu*

v i r t u a l vector =sons,

A very s i a l a r cheek giving precisefiy t h e s a m r e s u l t i s Ca we eq. f 18.1)

t o say A(yp

.+

YP)

° I

Again t h e d o d n a n t term 5 s t h e

p,

XndependatZy of the r e l a t i v e phases we

The q w a t i t i e e on the r i g h t h m d s i d e a r e available d i r e c t l y by e x p e r i m n t . Ihe dependence of dao/dt (yp

O f o .4 h V

2

-+

Vp) on s (from 2.7 t o 5.2 &V)

and on t (from

) a r e w e l l represented by t h e r i g h t hand s i d e , but i t i 8 always only

about 112 of t h e e x p e r h e n t a r c r o s s section!

Thus VDEl f a l l s here.

This is i n ka~cordw i t h the tiOX e r r o r i n the t o t a l c r a s s s e c t i o n t e x t , f a r the o p r i c a l theoreoln reLatee ilo/dt (yp square of a t

+

yp) i n the forward d i r e c t i o n t o t h e

(YP) (assuming s o wt h i n g about phases)

.

Vetor Mesons and Vector McwfrBorrrirrancrfH j p o t l t c ~ ~ ~ i s

How should the t o t a l cross s e c t i o n f o r y+ iducleus vary with the rnass We knew f o r eollisiclns with hadrons nuclear atarter

n u d e r of the nueleus A?

is nearly opaque ((because a hadxon-nucleon is c m a r a b l e t a the spacing of t h e nucleons) and t h e r e f o r e t h e c r o s s s e c t i o n f o r A nucleons is n o t the sunr of a contribution from each (hence otat

%

A o nucleon) because nucleons i n the

f r o n t shadow those i n back s o ultimately f o r l a r g e A. i t goes as t h e area; of the nucleus o r

.

On the. o t h e r band f o r photon-nucleus

c o l l i s i o n s a. f i r s t glance would

suggest t h a t the nucleus now being transparent fay nucleon is much smaller than nuclear spacing i n a nucleus) each nucleon would s e e the f u l l bean and hence t h e c r o s s s e c t i o n would go a s A ( t i m s t h e simple nucleon g h o t m cross s e c t i o n ) . Or,

t h e o t h e r hand t h e VLlM shows t h a t t h i s l a t t e r canclusion cannot be A photon m p l i t u &

r i g h t i n general f o r i t is i n c o r r e c t i f the V I M is c o r r e c t . should be proportional t o t h e p amplitude

- the l a t t e r

is a hadronic m l i t u d a

-

hence the photon cross s e c t i o n proportional t o t h e p cross s e c t i o n m d t h e r e f o r e varying aar A2f3

(F. neglect the? contributions of w and Q, i n t h i s q u a l i t a t i v e

d i s c m s i o n , t h e i r contributions a r e e a s i l y r e i m t a t e d ) . The reason is t h a t i n the s % y l e view we img1ned t h e i n t e r a c t i o n y

+ nuclem

= X t o be a l a c a l process of y on nucleon, t h e y i n t e r a c t h g a e a r

h e r e the nucleon is located can become v l r t u a l hadrons

- but VDM reminds us (e,g.

t h a t t h i s is n o t so.

A y

2n, but most i w o r t m t l y a p ) Tar away I n

f r o n t of the t a r g e t , and the, d r e w 1 hadrons propagate a long way t o t h e nucleon to interact,

A r e a l photon is a pure i d e a l photon plus a v i r t u a l hadron with

amplitude l/dE accord%ng t o perturbation a e o r y where bE is t h e m e r g y d i f f e r e n c e of t h e s t a t e of given momntm m @oton md as hadran, the energy Is very sml1 f o r l a r g e v.

%us i f i t is a p

while t h a t of t h e photon is v s o dP; %h'is

- vs

m 2/2\1, P is a l s o t h e distance ahead of t h e nucleon &ere

t h e photon-hadxan conversioa occurs,

m

I f t h i s is l a r g e coapared t o t h e m a n free,

path of p i n t h e nucleus then shadowing occurs (Azi3 1, i f i t is small, no s h a d w i n g (A), The physical i d e a can be seen b e s t f $ r s t f o r a simple mdr?l c m s i s t i n g of

two t h f a slaba a, b of nucfear ntatter me i n front of the other by d.

@e s h a l l

calculate according t o WH, d

I f a p m r e inrpinging suppose the smll p r o b d i l i t p of being absorbed

i n the f i r s t l a y e r is fa; i n b alone f f p iarp~i.n@d that the p gate t o b

is

(1-5) so the product

fb.

But &a probability

%+

lDade in b is ( 1 - 5 )

The

t o t a l cross section i s thua fa+(l-fa)% = f +f -f f the l a s t t e r n representing a b a b

L e t us ealculiare the prob&iIfty Ear any product8 lnade i n b i f a 2s present

by photone v i a VDM.

Xf a f e not presflnt the q l i t u d e t o find a p a t b

product i s proportfoaaf to the m p l f t u d e the y converts t o p a t x

('\/TF

(the

Y

:!&has3

) and theo the p a r r i v e s a t d.

Thi. Is

hhets been taken r e l a t i v e t o the photon phase a t x

.c

d I.e.

the

abwe expretsalon c o a t a i ~ satt addltfonaa! f a c t o r exp I-ik d ) ) where the k Y

clre the L-vector f o r y o r p a t the same frequency. large?

V (mk.

Y

Y.

- kP

m '/'v

P

'5 for

1*

When a is present (21.1) is valid an* the ad&tioaal factor f o r

x

Xence L Y

Q

for d > x

0, For x 1-fa/2.

t o get throulpk s =

c

O we have

%S f o r

O we have

B e total. m l l t u d e ( b e e g r a t e xf is therefore pxoportfonaf, t a f

The =in

-i(k -k )d P

(21.3)

t e r n has an m p l i t u d e proporrfanclf Cc?

m-------

Id

arpect, the term i n shedaring fa corms i n various ph$es

as=

- and i o a more complete

Vector Mesons and Fector Meson Dominance Hypothesis problem with a continurn of l a y e r s would cancel out f o r (k -k )d Y P

111

1.

Thus t h e e r i t e r i o n f o r s c a t t e r i n g involves v, shadwing should be canrplete

for

V

J.

"

One ~ z n extend t h i s e m f l y t o a t h i c k l a y e r i n one dirnrznsion.

WE? seek the

a q l i t u d e co fInd a p a t x {for t h a t v i l Z l a t e r be used by squaring and s w i n g m t o g e t t h e t o t a l cross section.

W e have two cases g 0:

Y

4

y converts a t

Y > 0:

y

O md y

>

O

Y, propagates ss

p to

0; propagates a s p in mdium t o x

c m v e r t s a t y, propagates in mdiw t o

X.

The aaplf tude is proportional t o

( t h e phases have been taken r e l a t i v e t o the photon phase a t

X,

i.e.

t h e above

expressions contain the a d d i t i o n a l f a c t o t sxp (-ik X)), V

I n t e g r a t i n g over Y the a w l i t u d e a t x is

Here k

P

si"

i n the mdiutn.

k

P

At very high

V

i t has an iaaginary p a r t (representing

absorption) which is f i n i t e and fixed, the, r e a l p a r t goes, of course with v, but

m y d i f f e r froan i t bp a f h i t t ? m w t ,

Thus a s

V

+ m,

k -k Y

P

'*

k a fixed

a d e r (rJhose inailg2nary p a r t gives t h e p absorption cross s e c t l a n and whose phase i s the phase of forward p nucleon s c a t t e r i n g ) . 2 Thus i f k -k = irouad

l700 and some fndication

Theoretically we should s e e more resonances, e.g. l535 but

of: a fourth a t l90Q.

undodtedly t h e 'kaeeond" peak is an unresolved d x t u r e of these twa while the one near 1700 s o m t i m s c a l l e d t h e 1688 resonance is probably t h a t with four others e w a c t e d near *at

energy,

Tne reaonmce a t 1407 ( t h e Rowr rasone~nce) has not been

seen i n these experimmts ,

can be f i t t e d m B r e i t - g i p e r peaks an a background whfch

These resonaxtees

gradually dominates i n t o a smooth curve .a :M

inereaaes.

How do the resmance s t r e n g t h s vary with q2? At verg low q

2

the behavior

depends t h e o r e t i c a l l y on the angular trowanturn of the s t a t e and @ t a r t s a s an appropriate p m e r of Q (gZ

2

-q ). Tor higher qZ, however, t h e s t r e n g t h of the

m s 0 m c : e s a l l f a l l more o r l e a s a s does the e l a s t i c peak. the r a t i o (do/&)

res

/(do/dQ)elastie

with -q2 oae find.

Zn f a c t i f one p l o t s

cvrvse which rise rapidly

from -ttzreshofd (photoelectric) values t o s a t u m t i m (In tnthe v i c i n i t y of one! o r 2 jut belou) f o r q2 > about 1 GeY

For l a r g e r q

2

.

and l a r g e v t h e r e is nothing l e f t of the reaonances, 2

i t wae suggested by Bjorken that t h e funcrion vW (q ,v) (and a L o Wl(q

2

2

2 -q /2Nv on3.y.

be a function of t h e v a r i a b l e x as a f a c t i o n of

u

llx).

2~v/-~'

This f e a t u r e , chatl m -q

true, 2

vM2 (q ,v)

2

+

approaches a f a c t i o n of

There ,v))

should

(Data La o f t e n presented a l s o

This has turned a u t t o be r a t h e r accurately 2 srsch t h a t -q 12M-v = x is b p t fixed,

W,

V +

X,

F(x) Is hm as B J o r b n scaling.

kle s h a l l

have a g r e a t d e a l t o say l a t e r of i t s t h e o r e t i c a l signiffcance. Present ( S a n u a ~1972) d a t a is a v a i l a b l e in the f o l l o a n g regicms.

Resonance h g i o n Scaling b g i o n Wl, W2 Separated

I n the region w regien 12

P

w

I n my v&ri&le.

4

4, s e a l i n g is observed above

4 f o r E$

,2

-pZ>

l; VW

2




-

a t reat)

Now t h e carnuttitor m t r i x element is defined by CBA(Z,l)

m

RgA(2, 1)

- Kh8(1*

2,

sa i t s Fourier t r m s f o m @ a t i s t i e s

We note t h a t X can be obtained from G and d e e versa because of ( 3 6 . 4 ) . %(v.

and CB*(-v,

4) -6)

-

cBA(v I

-

-CmLB(v,

6)

v >

Thus

o

(26' 6 )

6).

Thus 9.1% rar?asur&ng%(v,

w-tator

for

Q) ue a r e w a s u r i n g the Fourier trcrnsfom oE the

o f two c u r r e n t s ,

We s h a l l d-isccrss t h e cmsequenws a f t h i s i n t e r e s t i n g r e s d t i n a craomnt, but while we have these equations before u s we wish to derive a fczv Eomulm f o r the s c a t t e r i n g a q l i t u d e whlch we w i l l need l a t e r i n t h e w a r s @ , Aa we have

discussed the s c a t tering q l i t u d a f a r am inceaing photon ( v i r t u a l o r real) coupled t o J (1) say A(1) t o an outgoing one coupled t o B is detts-dned by the operator

we diacusa the e f f e c t of seaguljta ( i f there are any, 6 (2-X)

type terms) l a t e r

and omit them f o r a wbile,

If i n p a r t i c u l a r we are intezleclted i n the Eomard s c a t t e r i n g aw1itude TBA(4 00" a proton with a photon of mmnturn q we need the Pourier t r a a a f o m

(The superscrfipt F indgcates- t h a t the choice of

siw

of the i m g i n a r y part

f o r negatfve frequencies is taken accordgag t o the convention of Fewmn i n h i s

QED papers; there is a dif f e r e a t choice called u w a l amplitude.

which is

often very useful,) To take the Fourier transfofln of the f i r s t t e r n i n ( 3 6 . 8 ) we have a product o f @(t2-tl) & w e P.T.

i s i/(v+ic) and B(Z)A(1)

we have the convolutlian of these,

whose P.T. i s %A(v,

P

use

&t

hence

In the s m e way the a e c ~ n dterm is the

convolution of -%/(v-%E) (the F.T. of @(tl-tZ) and Km(-v,

NOW

6) and

4).)

Hence

inscvr-v>

where we have exp1icitl-y added possible seagull t e r n aBA which L s sinply an u n k n m ( f i n i t e order) polpemial i n v,

8, Thus

the s c a t t e r i n g m p l i t u d e is,

except: f o r a golynmial, ESfven i n t e r n of the comutator, Megatiw

V,

58 of course, not defined by experimnt,

Hawever i t can be

obcdned from nnssasuremnts of the reaction with a n t i p a r t i c l e s (using

A

f o r A)

v i a the connection implied by (36 ,IQ) m d the r e l a t i o n f o r comutatars ( r e s u l t i n g

f nelasric Scattering as Propertl'es of Operators from the f a c t t h a t

a))* -

(c,,(,

is the adjoint operator t o A)

a, - -c,

(V,

41

(-V.

so t h a t (36,101 i m l l e s

F

TBA(v.

6)

-

T&(-w*

41

h o t h e r more- usual convention (causal a ~ l i t u h )t o define the negative frequency exteasion changes the a i m of the imaggnary p a r t f o r negatlve frequencies

a, -

T;~(~,

T;~(~,

*

C

Thus (TgA(v, 9 ) )

-

a, +

c

CBA(~.

.

4)

T s (-v.

and a l s o (37.1) beeoms n m

(vhlch is j u s t as good a way af describing s c a t t e r i n g

The s i p i f i e a n e e o f F

a s T ) i n coordinate space can a m be seen.

e

(=I

given by an expression 1ik

( 3 6 . 9 ) except t h a t the sign of the +ic i n the l a s t term I s reversed,) In (37 .l) we have expressed

things i a terols of the carnutator but the l a s t t e r n caa be

m r e @%=plyexpressed a s the product operator, from (36.5) and (36.6) we have

6)

@(-v) v,(C ,,

-irC

-

-

-KABf-v,

41,the

F.T. (B(2)A(1)IT

f o r t2 2 tX,we have B(P)A(1) A(1)8(2)

- A(I)B(Z)

P.T. of eA(1)B(Z).

Hence (37.1) says

- FeT.A(1)B(2)

- A(l)B(Zf

P

[BQ2), A(1) 1 and f o r t2

have

1.0

. By retarded e m u r a t o r f o r t2 C t

c tl"V

we mean the c o w t a t o r

1:

froar. t h i s , (37.2) i s d i r e c t l y obvious. (Rmrk,

The (3(t -t ) i n 37.3 a t f i r s t s i g h t &es the r e s u l t not relet2 1 t i v i s t i e a l l y invariant, u n t i l i t f a realized t h a t the comutator Eacmr is zero Ear a p a c e l i u regions so the p l a e t2

t1 c= be t i l t e d a r b i t r a r f l y as requrfreci

172

Photon-Hrzdron Interactions

by Lorentz transformation.

This i s true a t lease i f Eke comutatox is not too

singular a t equal times which

it9

often true,

T i A defined here not r e l a t i v i e t i c a l l y

MEficulties s o m t i w a a r i s e A i n g

isvariant unless corresponding non-invar$ant

terms (called Schwinger t e r n ) a r e added i n t o the '%eagullif part of (37.3) .) Expression (37.3) serves as a general definition of the Chew amplitude, i . e , i n space-tinart lit is the retarded comutbitor, even when we do not have the diagonal e l e m n t a , or lowest s t a t e s (so the product operator is O f o r v

0).

The general definlltion of the Feyman ampiftude is the time ordered operator ( 3 6 . 8 ) . They d i f f e r by the product operator.

Note on various refations b a l l t y conditions

Crossing C

6)I *

fTM (v,

-

-

a, T',

P

T ~ ~ ( ~ .

-6)

T&

(-V,

Imginary Part G

iTa

(V,

a))* -

4,

a, - -~G,(V, a)

~,(T;

(37.8)

The crBA s a t i e f y r e a l i t y and croiilsiag relations required t o keep then

valid f a r T, GM*

(v,

Nmlg

41

m

G Z

(-\?p

-4)

-

0 s (v,

The commutator a t equal times t2

vlth respect t o v f o r a l l v, since equal t i m c ~ m u t a ton i relation

61

tl can be obtained by integrating CBA(v,

e" (t2-rl)dv/2n

6 (t2-t l).

6)

Hence Cell ELann 's

173

Inelastic Scartorirzg as Properties of Operafors becomes upon Fourier t r a n s f o m f o r our diagonal element on t h e proton, a,

a constant, Independent of

6,

This is c a l l e d a s u s r u l e .

We now r e t u r n t o our study of the f a c t t h a t i n xneaeurlag R

BA (v, Q) we a r e

m a s u r i n g t h e F,T,

of t h e c o m u t a t o r of two c u r r e n t s ,

we s h a l l ask a r e ( l ) Mhat limitations on the F.T.

As examples of q m s t i o n s

r e s u l t from t h e f a c t t h a t t h e

c o m u t a t o r vanishes outeide the l i g h t cone?

12) From e x p e r i m e n t ~ lf a c t s about

the behaviar of IZ ( e .g, Bjorken s c a l i n g ) &at

do we Ittarn about the character

of t h e c o m u t a t o r ? It behooves us t o study the general behavior of c o m u t a t o r s , and we begin

our study with the c o m u t a e a r of two a c a l a r f i e l d s of mss nn i n a syhstem without

We can express 4C1) i n t e r n s of c r e a t i o n a d m n l h i l a t i o n operators i a the usual way

Were

W

k

=

is the c o r r e c t frequency t o describe the o p e r a t o r ' s

developmat in time f o r t h e r e 1s no i n t e r a c t i m s o the enerC?;y i e ; t h a t of a f r e e particle.

The

commute with each o t h e r , and the %*'S, only a and a* do n o t

c o m t e i f they belong t o the s m e %:

We work out (37.21)

i m e d i a t e l y by fomhlng t h e c o m u t a t o r , w i n g (37.131, of

an expression l i k e (37.12).

Dropping t e r n which obviaualy c o m u t e we a r e l e f t

with:

The a c t u a l i n t e g r a l can now be done

- i t invobea

Here we w i l l j u s t do t h e s p c i a l caae m

-

Bessel functions,

0, and say what the case 'm

$ 0

174

Phorsn-Hadran Inreractions

gives, l e w i a g d e t a i l s t o the st-nt,

-

In (37.14) put

&OS@,

%

-

L for m

-

O t o get

singular 6 f w c t i o n on the 1i&t cone. For f i n i t e

we obtain &e

s h g u f a r i t y on the ligttt cone, zere

outside i t of coursle, aab a Bessel Emetion inside:

Thus we see t h a t the c-utator

f o r f r e e part?iclea is zero outside the l i g h t

c m @ and singular ( l i k e 6 f m c t i o n ) m the l i g h t cone, We helve discussed the singularit2ea of two f i e l d s , dil~cussthe ~ J n g u X a r i t i e ef r m tnuo currents. f i e l d theory J (1)

(2p+ ap)# ( l )

Also i n s t r u c t i v e i$ t a

Qne suck current ~ g h br e i n a free

@ ( l ) f o r spinor f ?ields. M an e x a w l e we w i l l leave out tlzeee gradients, e t c . asld fgnd the c-utator P

6(1)

or

h

for

"currentsW"tat a r e simply squares of a s c a l a r f i e l d

where s ( 2 , l ) i o the f r e e EgeeM c o m a t a t o r we worked out before. 2

has the s a m s i n g u l a r i t y sgn (t2-tl)6(52)

Clearly K

8s a ~ t c dbefore f o r C, but thts t i m

tnu1t;lplied bp an operator function of two pcsitions F f 2 , l ) called often a b i l o c s l operator,

This operator is needed, of course, only m the l i g h t cone.

Its matrix elements give functions of xZY-xlli

( f o r e x q l e of t2-tl f o r a

dtagonal elemnt: i n a system of a partlclie a t r e s t ) .

So i n g e w r a l the

sinlngularities along the cane a r e raodaated by a f m c t i o n of the distance from the origin of the cane, Far interactgag hadrons the e m u t a t o r of currents outeide the liglnt cone also, and non-zero only i w i d e . h w i t &as

the t r ~ s i t i o na s we cross the l i g h t cone,

ir,

elrpected to be; zero

An i n t e r e s t i n g question is

So-

s o r t of jump Is

i n slope? Or possibly i t has 6

Is i t i n value or on*

expected p r e s m b l y .

Such a question is a f m d m n t a l one

function a s the f r e e p a r t i c l e case does.

Me have the Fourier t r m a f o m e x p e r i w n t a l l y i n K

UV

(v,

$3

, B. olngular

behavior corresponds t o so= s o r t of high v, high 4 l l m i t ; so i t is the b e h a d o r a t large v, Q t h a t gives us the answer t o the question. experiaents indicate R(v, Q) s a t i s f i e s BJorken scaling function of 6

But i n t h i s region

- t h a t Is,it

58 only a

2

-q /2&,

The most otxaightfomard way t o a a l y z e t h i s is t o take the inverse t r a n s f o m of K, using the BJarken l i m i t aad aeeisrg what s l n g u l a r i t i e s i t g i m s

- then

concluding the character of the s i n g u l s r i t i e s , because they depend only on the high v, Q I i d r a r e the ame f o r the c m p l e t e function as they a r e f o r the Bjorkcm X i r n i t .

We wZl2 not t r y t o be rigorous, becauee f o r oae thllng we r e a l l y

do not know, experimentally, R f o r q2 Let Q be along the z axis.

h r g s v , QZ is nearly

V

-q2

p

0, nevertheless we can see what happens. ~

i n f a c t qZ = *MC.

~

(q>v)(qz-v) ~ v

ZMvE, ~ hence f o r

The Fourier inverse of e f m c t i o n

o f F, only would look l i k e

where s ( t ) i s the F,T, of the s t r u c t u r e f u t c t i o n f ( F ) , To put i t in anather s l i g h t l y m r e rigorous way we nors

Now omit f o r

41

xaOfDent the 2Hu (suppose we took the txmsforn of M ) and

a

note f a r l ~ r g ev, qa t h a t t o a good enough approltimatten the 6(q 2+ZWvB) can be replaced by 6 ( q 2 + 2 ~ v ~ + d ~ 2So ) . we get f o r large enough v, (q2+ Z M ~ B + ~=B(q+@p12) '

Now we can i n s e r t , i f ve. wlsh, a sgn(v) t o keep the as f o r a F.T. cf a c o r n t a t o r , m d f o r large v t h l s is j u s t sga(*BH] proton is a t r e s t ) .

Hence a s p p t o t i c a l f y

( i f the

1 76

NW

Phutoa-Hadrm Interactions

the F.T. i s easy f o r we know t h a t F.T.

sgn(t)d(s

2

2 sp(vId(q I

s o that f o r a four vector a

hence P,T.W2

noting P*x

- jf

( ~ ) d 8e-iBP*xapn(t)6(t

2

2

-R )

MC we have our previous r e s u l t

- t h a t the significance of Bjorken

ecaling is t h a t the s h g d a r i t y of the current operators has s i n g u l a r i t y on the l i g h t cone.

a,

6 fuaction-like

To be =re precise we w i l l have t o include all.

&e P P factors, etc., and define everything precisely U v gradients of 6-fwctions involved,

- there w i l l then be

The beat way t o eay i t i n a general way is

t h a t scaling shows t h a t the sfslgularitfee on the l i g h t cone of the current carnutatore a r e J w t of the s a m severity as they would be f o r f r e e p a r t i c l e fields, m i a is, of course, what we expect from the parton interpretation f o r there we a s s w d I n the f f n a l stateax(vhich beeme tlhe h t e m d i a t e s t a t e s of the c m a t a t o r ) the partons can be consfdered a@ free,

Thus there Is no surprise

i n the conclusion, i t is m l y t h a t we whLElh t o s t a t e what we have discussed (scaling) in an a b s t r a c t and general wag (as a r r a n t comatator sialngularities) with m%nimm references t a a model.

Although the s t a t e m n t s t i l l s e e m t o r e f e r

t o f r e e p a r t i c l e f i e l d s , that i a only a shorthand t o writing 8 and 8 ' functions on the l i g h t cone, Each general property of partons we assmed s a i d smathing m x e e x p l i c i t about the character of the singularity.

For exmple, t~ r~aycharged partow

are spin 112 i e t o say the a i n g u l a r i t i e s are l i k e those 6 o r t e r i s t i c of f r e e Dirac f i e l d comutatoro, r e s u l t s a r e related.

ffunctions charac-

Ea addition, vector and iai& vector

I f , f o r exanple, one adds t h a t partona a r e quarks c e r t a i n

n m r i c a l l r e l a t i a n s are implied, aa we have seen, between the s f n g d a r p a r t s of the comutators of variaus klzlds of currents.

( m a t i s t o say f o r examle, our

Inehtie Scattering as Propertiw of Operators r e l a t i o n s among feP and f i p , f i P e t c . implied by expressing everything i n terms of the s i x f m c t i o n s

U,

z, d , 3, s, z,)

Quarks, a s f r e e p a r r i c l e s , have n o t been fowd. a s t o whether the & t a i l e d views sf tbe partan =del

There a r e many questions a r e correct f o r quarks.

In

p a r t i c u l a r , the question whether a s i n g l e parton quark moving out i n recoil, need

o r need not rake i t s non-integral qwatum a m b e r with i t . In addition i t i s o f t e n very useful t o aee, vllen r e s u l t s bave been obtelned from a traodel, J u s t how mu& depends an the rnodel and whether i n f a c t the r e s u l t s carnot be s t a t e d aa a general mathematical p r i n c i p l e without: recourse t o a s p e c i f i c model, =del

In t h a t way, i f i t p r o w s l a t e r t h a t too a m y d e t a i l s of the

a r e f a u l t y , we can r t l l l begin again having learned s o w general properties

of hadrons w%thout being c a m i t t e d t o a11 t h e o t h e r s , m e r e f o r e S t i s i n t e r e s t i n g t o note t h a t t h e parton m&l is equivalent t o the general statement: (and one mueh c l o s e r t o d i r e c t elrperimental r e s u l t s ) t h a t c m u t a t o r s have 6-IXLe Zigbc-coae s i n g u l a r i t i e s

.

Light Cone A

Lecture 38

AR I n t e r e s t i n g qt1138tion mswered by Britzsch i3nd a l l - m n (1971 Coral

Gables conferenca)

Ss haw t h e s c a l i n g r e s u l t s of t h e "partons as quarks" c m

be s t a t e d i n a way wbf ch

- at the end

doesn't involve quark wave f m c t i c n s

o r operators a t a l l , We wish t o s t a t e t h a t t h e Sight cone a i n g u l s r i t i e s a r e l i k e those f a r f r e e

quark (spin 112, SU t r i p l e t ) comutatorB. 3

We man, of course, t h e l a r s s t

6' o r d type of s i a g t t l s r i t p , as we say t h e '%leadingw ( i n high frequency)

singdarity,

F i r s t we s e e what t h e s i n g u l a r i t i e s a r e libre 2 f currents were

represented by quark f i e l d s * ( X ) ,

a M r a e s p i n o r carrygag SUS i n d i c e s on which

3 x 3 rrratriees h can act, a"he conanuratror o f two splnar f i e l d s of m s m is e e s i l y worked o u t (as we did f o r Base f i e l d s ) i t is

( J u s t ss t h e propagator i s

+m

(f + m) /(p2

- m21 i n s t e a d of

l/(pZ

-

2

SB

1 so this

c o w s l i k e v i s e i n t o the commutator ( t h e r e a l p a r t of t h e propagator).

Since we a r e looking f o r the leading s i n g a a r I t y near the l f e t cone Gm Is.

Light Cone A fgebra J(+

2

and the w s t e r n i s

s2t

where

w l l e r than the large gradient, thus

is the square i n t e r v a l from 1 t o 2 and 2 means "the leading

slnguLaritfes near the l i g h t cone are equal. '' It i s e a s i l y v e r i f i e d t h a t the- r e s u l t s we: obtain here f o r the c m a t a t o r s

of two currents each a b i l i n e a r form i n

v,

a m exactly tk s a m ðer we

suppose s ) the f i e l d s obey the usual a n t i c o m t a t l o n r e l a t i o n s a t equal, e l m @

--

T(%)$(i;2)

Y ( x ~ ) # ( x ~+)

-

- Z2) and +(Z1),+(Z2) anticommute a s i s

&'(gl

appropriate t o Fern2 p a r t i c l e s ; o r we suppose b) the spin 1112 f i e l d s abey 3 c-utation r e l a t i o n s ~(%)y(;~) F(xl) +(x2) 6 (g2 and

-

-

cornate

eu3

- 5)

+(z2)

is approprgate t o Bose p a r t f c l e s ,

Thfs i s very Iuterleratl-ng becawa i t says the ""Bose qusrrk" "del

whfch

is appropriate a t low energies i a f a no fundmental way %a contradicttlion t o '"partens

as quarlcsl%t higher enerm.

a8 808e quarks

We

can call t h i s partan ~ltoclel""prtonsr

."

Next a current of SUS type a (deseribedw ha) i s

where f o r e x a w l e f o r e l e c t r i c current h" Is diagonal say hY

-3 ) .

diag.(2/3,

- l/>,

For axial current8 change y p t o

NW when we comute

currente,

WO

r e s u l t (see reference f o r detagla),

find a s h p l i e but tedFousZy complicated

W

Tc i l h s c r a t e the idea, r-Jhich i s a l l we

intend t o do here, we t&e the ease of two electrFc curreate, and also drap several t e r n s which would vanish i f w e took s p i a averaged matrix e h m n t a , then a l l that aumives i a f ~ : ( 2 ) , $:(l)

2

Bp

(

E

(

~

~

-

l

~

~

)

~

(

~

-

,P-&

~ ~s t (~2 . 1~ ) ~ 1 aT ( 1~~ 2~1) Y ~ Y ~ Y (38.3)

(plug other t e m a dropped)

(2,l on ntutwl l i g h t cone) and h

6

is diag, (4/9, 1/9, 119)

hYhY i n our caeee.

Mow we s t i l l see the quark f i e l d s in ( 3 8 . 4 1 , but the Idea n w i s t o dis-

regard

equation (38.4) and t o suggeet t h a t e q w t i o n (38.3) ( i t s generalization

t o a r b i t r a r y currents and i n c h d f n g ola;ttted t e r n of c o ~ r s e )a r e geaerally valid.

They give the l i g h t cone singuXarities and define, on the l i g h t cone a t l e a s t

.

a s e t of new operators Va(2-1).

It Ss t h e matrix elements of these operators

which give us our s t r u c t u r e functions. No d i r e c t inrplication is m d e t h a t they can be expressed a s (38.4). Eqwtion (38.4) as well a s (38.11, (38.2) a r e j u s t s c a f f o l d i n g t o a r r i v e a t (38.3) and a r e henceforth t o be forgotten, quark operators a r e seen.

I n (38.3) no e x p l i c i t

Instead only soae new b i l o c a l operators a r e defined

(mctanfng depending on two points).

They a r e defined only on t h e l i g h t cone by

the very equations (38.3) f\

[This system would be a t r u e "algebra" i f the p r o p e r t i e s of the ,J" could now be defined independeatly,

For e x a m l e , an ideal. s i t u a t i o n ( i n f a c t a

complete theory of t h e hadrans) would r e s u l t i f equations giving the Comutatars of such

ps

could be given i n t e r n of J% and

seem possible,

A l i t t l e can be done, hcwever.

?'S

themselves,

T h i s does not

Wfth the fonn (38.4) t h e

R

e o m u t a t o r s of two :g's which have t h e i r v a r i a b l e s on t h e s a m l i g h t ray (e.ge ~ : 2 ( 3 , 4 ) f*'(l,i?)] ,

3,4,1,2

*ere

cone) again can be expressed a s m but i t is n o t much. reactions l i k e e

+p

a l l l i e on a s i n g l e generator of a l i g h t This r e l a t i o n has been a l s o hypothesized,

E t gives p r e d i c t i o n s f o r c e r t a i n two-current i n c l u s i v e -c

e

+

p+

+ p- + any

those expected fro= t h e garton =del

hadrans.

They a r e , of csurse, J u s t

i n t h e s a m conditions,

Me discuss them

later,]

I n order t o w e equation (38.3) we must take the Fourier t r a n s f o m , and f o r t h a t we can w e t h e r e s u l t (very s i m i l a r t o those we have already derived, we leave d e t a i l s t o you) i f q

(v, 0, 0, u 4- N Q

t h a t is, i t involves only t h e i n t e g r a l along t h e ray t = z camon t o t h e plane t = e(from eiv(t-z))

a d t h e l i g h t cone.

Naturally every r e s u r t of t h i s theory i s a l s o a r e s u l t of t h e "parton as quarkw "eory

f o r t h e l a t t e r is a model of t h e f a m e r ; but n o t every parton.

r e s u l t can be derived from (38,3),

I t i s possible, therefore, t h a t =any of

them a r e wrong and only (38.3) and n o t the conrplete f i e l d model may survive.

Light Cane Algebra Therefore i t is i n t e r e s t i n g t o compare the t h e o r i e s f o r varfous types of predictions raade by t h e parton a s quarks theory,

They seem t o be of t h r e e

classes. A.

Scaling, r e l a t f a n s among scalfslg functions, sum rules.

B,

Sgecial a r g m e a t a about the s t r u c t u r e futlctions derived by a r g u m n t s about hadron c o l l i s i o n s and o t h e r a r g w n t s ; l i k e dx/x behavior, t o x near l, s t r u c t u r e function

r e l a t i o n of form f a c t o r power

{I.-XI~ e t c . C, Applications t o experiments of D r e l l type e+

+P p -t any

+ e-

+

hadrans o r

hadrms, etc. p + p + Glass A a r e e x a c t l y those derlved from Sight-cone algebra, obtained from l i g h t cone algebra by d e f a u l t , discussion of how t h e a a t r i x elemeat of '?{2,l)

Class B a r e aoC

That is, they simply m o u n t t o a mtght behave,

X I is n o t a

s p e c i f i c a s s m p t i o n o f the m d e l but an attempt t o go f u r t h e r t h a t leads t o Class B r e s u l t s

-a

s e r i o u s attefDpt t o discuss the nratrix elements of t h e l i g h t

corn algebra would lead fo r e s u l t s in t h i s c l a s s . Class C a r e very i n t e r e s t i n g , a s they seem beyond (38,3) and require so= extension, even thou*

they appear evident from t h e perton view.

It f e therefore

here, t e s t i n g these t h a t a r e a l choice can be made a s t o ðer the extensions of the parton model beyond t h e expectationcl of (38.3) a r e r e a l l y sound, Tlze reason the c u r r e n t c o m a t a t o r oa t h e l i g h t cone is n o t s u f f i c i e n t f o r reactions l i k e p

+p

+

p

+ + U- + X

is t h a t we do need m t r i x eletaents l i k e

but t h i s time a s we take t h e l i r a i t

M

the anomntum of t h e c u r r e n t operator

increases we a r e a l s o changing t h e s t a t e pp; t h e r e l a t i v e mamentm of t h e two protons mwt a l a o increase,

This f e an a w h a r d l i m i t f o r a theory of operators.

It is a very i n t e r e s t i n g question,

1s t h e r e so= general a b s t r a c t way

without quark f i e l d s (partons) t o describe a l l these Class C parton p r e d i c t i o n s also?

Or a r e they perhaps wrong, m r e l i a b l e extensions of t h e idea?

And what becomes of the questian of how the partons c m @ a p a r t i n t h e proton without exhibiting quark qwntura nu&ers m n g the f i n a l s t a t e s ?

Does

current algebra h e l p us t o solve i t ? Perhaps, i t s e e m t o be t r a m l a t e d i n t o "are there any representations possible of t h e algebra (38.3) &ich do not imply quark quaatuta nu&ers atoong t h e localized s t a t e s ? t h i s anathemtiral f o m than i n physical a r g m n t s .

It may be m r e t r a c t a b l e i n

Properties of Commutators in Momentum Space

Lecture 38

We now cm@t o discuss what general properties a carnutator has t o have i n Hlomnturn space, such t h a t its Fourier trslneforrn w i l l be zero outside the l i g h t cane of configuration space. This is c l e a r l y a prolnising l i n e t o follow t o a u p p k m a t physical, i n t u i t i o n on the properties of Wvv.

At present not a great deal can be s a i d , but f o r your

i n r e r e s t i a your future research &at has been done here m y prove f m i t f u l ,

First we can

so-

obvlous r e m r k s ,

I f we multiply a comutator ( i n

space-tim) by a function G which is 1 inside sad zero outside a l i g k t cone we recover the w m u t a t a r (except, a s is the case, t h a t it is s i n g d a r right m the l @ t cone where our funct%on G is poorly defined). isfi

2 2 l6aP,V,(l/q ) (O.V.

Since the F.T.

of G

pxincfpal value) we have the convolution theoran

( p l w piece@ from the 1igbr cone). (C(q) is the F,T, of a carnutator,) Xn a similar vein, since the F,T, of we see that G mwt b v e the form

182

O

is f b n i sgn(qio)a'(q

2

183

Properties cif Commutatorsin Momentum Space

O f course

plus l i g h t cone pieces,

.wr?

can say a l s o i n (39.1) t h a t the C(u1 sf

the iategrautd is any function F(u) a t a l l , and the C t h a t c o w s out on the l e f t w i l l be the F.%',

o f a ftlnetim zero outside the l i g h t cone,

1 have n a t been

much use aE these observatiom.

able t o m&

I n using the simple views h o v e we awl: be careful of one paint.

'Z"he

f u n c t i ~ nwe multiplied Gix,tZ by, G ( x , t ) , wars &fined as l i n s i d e the l i g h t cone

and O outside; h a t i s i t exactly on the l i g b t cow? ordinarily that is such a fsrnalli region t h a t i t &as

T t i s i n d e f i d t e there, no difference in the i n t e g r a l

over space tiw of C ( x , t ) G(x,r) but i n f a c t i t make@ a great mcextaiaty because Gfx,tlhas a 6 function s i n g u l a r i t y j u s t &ere G(x,t) is poorly defined.

We can

correctly straightctn out our Eomulas f o r t h i s e f f e c t i n the following way,

kle

c a l l Ca(q) the a e m p t a r i c C(q) g i v h g just the light-cone 6 s i n g u l a r i t i e s ,

man

C(q)-Ga(q)

has no l i g h t cone s i n g u l a r i t y m d eo equat20n 139.1) f o r e x a w l e , holds

i f G is replaced by C-C,

on both eridee.

Qne c m then s i q l i f y o r rearrmge tfie

equation t a obtain one flke (39.2) but with s a m ~ d d i t i o n a lt e r n r e l a t e d t o the litght cone beethaiviar, A f sr more s u b t l e and useful observation was ~nzadeby Dyson,

i n m a y probless we a l s a know C(qf

He noticed,

O f a r c e r t a i n regions of q space (where

oo incermodiate s t a r a a may be available).

For example, f o r qO

v > O the

lawest s t a t e avall&lle f o r the f i a a l s t a t e x is the pratan i t s e l f , aroving with lasoaentlzla Q (space p a r t a t q ) hence with energy

v

hen-

and f i b w i s e v < + H -

*

&son he@ proved that the necessary a d suf f i c i e n t condf tioton f o r G(q)

veniah i n repion S

S1(q)

to

q0 < S2(q) and t o have a Povrier transform vanish

oufsi4e the l i @ t cone i a t b s t C(q) cam be written tldt

where @ vanfbshr?cr outside a regiaa R, but i s othbtmlsrst a r b i t r a r y ,

The regllon R

i s such t h a t t h e (q-u)2-s2

hyperboloid does not penetrate region S.

That i t i s a s u f f i c i e n t conditian i s easy t o see ( t h e g r e a t d i f f i c u l t y of t h e proof i s t o show i t i s necessary),

We have already seen t h a t the f r e e

p a r t i c l e c o m u t a t o r Cm(x,t) is zero a u t s i d e the l i g h t cone, s o t h e F,T. 2

2

1

(q -m 1 s p ( q

is zero o u t s i d e the l i g h t cone,

by anything, i n parCicular by P

iu-X

But i f C,,(x,t)

of

is m l t i p l i e d

( o r any superposition over U) It i s s t i l l

zero o u t s i d e the l i g h t cone, hence F.T. eiue"~m(n,f)

sgn(qo-uo ) 6 ( ( q - ~ 1 2 - d ) . 2 2 Hence superposing with weight @ ( u , s ) various cases of u and m s2 we g e t

I t i~ easy t o see, by drawing hypczrbolzls t h a t f o r our caee t h e region R

2 o f i n t e g r a t i o n where @(u,s ) does n o t vanish i s a s follows

i n s i d e two canes For a2 >

2

For s2

M!

h0k/4

U

i n s f de region bounded by M -lug/

-

u

h c i d e n t a l f y , t h e s c a t t e r i n g a q f i t u d e t h a t corresponds t o t h i s c m u t a t o r ( w i n g 337.2) becorns

where o (from s e a g u l l s ) is a polynomial i n q u ( f o r the amplitude

replace

LE(qo-aO) by i h ) . One of the d i f f i c u l t i e s I n w i n g the Dyeon representation is t h a t the function @

is n o t unique, m n y can give t h e s a m G(q).

has Been derived (I suspect not: aa rigorously

Another very s i m i l a r representatian asj

Dyaon's) by b s e r , C i f b e r t and

Sudarshm e s p e c i a l l y f o r a problern l i k e eure where C is a function only of t h e two i n v a r i a n t s

QL,

v,

i t is

185

Propertb of Commutatorsin &mefiturn S p c e This is, of course, j u s t Dyeon's representation i f the four vector

U

can be

a s s m d t o have only a time component, t h a t is, only a contponent i n t h e p

P

df r e c t i o n . There is again the expectation t h a t B(a,B) is zero o u t s i d e a more l i m i t e d

-G/Mt o

Leave our expressions i n t a c t , j u s t rentexnbexing If(a,B)

6 for

6

2 H

.

2 M , B only runs from

range i f o

In f a c t i f a

.t.

&/M, but we in

this region of a .

L t is c2ear t h a t if C is zero on t h e l i g h t cone various gradients a r e s o s e v e r a l p o s s i b i l i t i e s e x i s t with d i f f e r e n t i n t e g r a l p w e r s of v [from

&so,

zero up t o any f i n i t e v a l m ) with correrrpanding B functions; h u t we have taken the one appropriate t o 2Wl.

Equivalently we c m w r i t e our f o m with a new

definition f a r o,

I n addition, f o r our c m e , t h e a s y m e t r y i n

tlrtnas

f o r the e o m u t a t o r becorns

t h e property t h a t

T h t ~representation is very nice, and h is prob&ly tyQiq2y deter&ned by HI.

The weaicnera, h w e e r , i s t h a t no physical i n t e r p r e t a t i o n o r ezrpressfon

i n t e m of nratrix e l e m n t s is given f o r h(cr,@), T'hereEore we cannot use any physical i n t u i t i o n I n g w s f r h g haw h should behave? (other than using our h w l e d g e of how M brsheves and trorking bakward), @ay

- such a f w c t i o n f o r h(a,f3)

4.8

m a t 8s t o say we c a m o t a t any s t a g e

too 'krazy" "yeically

behave s o and fro i n t h i s mgion, e t c .

- or

i t ought ta

As we s h a l l see, t h i s i a a s e r i o u s

weakaess i n t h i s carse. We am t u r n t o s e e how h(a,B)

=WC

behave In order t o produce a function

2MWl behavlng a s we expect (see UcCWcs 3l)for l a r g e

F i r s t of a11 we have t h e s e a l i n g l%&t

Bere

= f ( X ) a Emetion of

X

only.

v

+ m.

Eq. (39.7)

V

(we t a b v

2

-q

givemt

O throughout).

.r m, - q 2 / 2 ~ v

-

x finite.

(sgn(v+BM) = I f o r

large v, s i n c e v > M).

Now i f we a s s u m b(a ,B) lnvcalved here, so

v

f a l l s away rapidly enouejrt with a only f i n i t e

c;

are

the

a c m be dropped i n the 6 function and we have

X believe i t was t h i s a r g

t with t h e Dyson retpresentacican which e i t h e r

+

OD

l e d Bjorken t o Iris s c a l i n g hypothesis, o r helped t o confilaa h i e e w p i c i o n e of

its t r u t h , qL >

o

scaling

We have here a ISOR=+ should go i n the p o s i t i v e q 4-q

2

jJe:

2

2

c m n w deternine how the f m c t i o n 2M1(q r v )

s e a l i n g region.

Letting v

+ m,

q2

-r

such t h a t

m

/22Vfv = x' is f f n i t e we have 2Wl =

(D, -xt))da which again s c a l w ; but

even m r e , from the s

NW c m we s e e t h i s rernarkablle r e s u l t from the partan model? equally well derive t h e funetion f o r qZ

; .

Can we

0 by using physical a r g u m ~ t s l We

w i l l explain q u a l i t a t i v e l y haw P t corns etbout. Por p o s i t i v e q2 we muat be concerned i n the c o m a f a t o r (36.1) with the 2 subtracted v a c u w p i e m , I n the vacurn we eaa a e p a i r s , and a t high q jwt

kle a r e concerned with hcrw t h i s p a i r production probability is tllodified by t h e presence of a proton

At f i r s t one oofght think the niodification must c o w throwgIr i n t e r a c t i o n of

187

Proparties of Commutaforsin Momentum Space the newly fnade paxtons with those i n the proton

- in

i n t e r a c t i o n about which

we know l i t t l e except t h a t i t is f i n i t e and i a not involved i n our derivation o f f(x) f o r negative q

2

exclueion principle

.

But a t high energy a: =re important effect: Ss the

- partons c a n o t be m d e i f

the proton ( i f F e d s t a t i a t i w is a s s m d ) . b i l i t y t o produce a the charge is 2/31,

Thus i n our diagram the proba-

t o the l e f t mid u t o the r i g h t a t x ' i s 4 / h U i t s (for Hmever, i t is a l t e r e d a t those x by the chance, u(x), that:

a u parton is present i n the protan. t o (4/9u(x9; i f the

they a r e already pre8ent Sn

G

Thus we have a contribution p r o p o r t i o n d

parton is w v i n g fomard i t is (4/13)~(xV. ZE a d par-

ton p a i r is produeed the probability Ss 119, mdifsed by the chance af f i n a n g

a d partoa already i n the proton, the contribution i s (1/9)d(xV, e t c . ) the e n t i r e contribution is 4/9[u(x8 f&(x")l+

1/9[d(x')+d"(xt) )l

Thus

+ 1/53 [a(x ")+;(X"

)l

o r fep(xt) ss required.

I n =king atstisties,

2

the analysis f o r q positive we aae-d

the quarks obeyad F e d

I f Bose s t a t i s t i c s a r e used i n the f o m l expression of the con-

pLeted surn over s t a t e s (34.1) we kave aoae changes,

F i r s t there is a change i n

s i g n of the F e w exclueion e f f e c t turning i n t o a Base tendency t o increase m i s s i o n beeawe a p a r t i c l e is already present,

sip

iin

men there is =other

chage in

the closed-loop diagram 4 [JJIO> when changing f rorn F e d t o Boae

s t a t i e t S e s [see, f o r exmple, R,P, Fe

, Phys,

756 (1949) 1,

Rev.

On the other Ixmd with the Botle s t a t i s t i c s sign the iaagiaary p a r t of

< O ~ J J ~ Ois> negative and i t cmnat be written ae a sum of positive p r o b a b i l i t i e s C l l

2

, but

1s minus t h a t

SW.

'Fais circurnstmce i s the b w i a of the

X

w u a l proofs of the r e l a t i o n of s p i n md ~ l t a t i s t i e s ;spin one-half p a r t i c l e s c m o t be consftltently gncerpreted ss Bose p a r t i c l a s , incerpre t a t i o n o f Base qmrlcs would

e e 3.

-I.

l%e mst s t r a i g h t f o w a r d

the "'Dxelf Constant" defined f o r the

any hctdrons ctose section waufd have the impassible value -21 3 ,

a%

course a c o w l e t e l y naive i n t e r p r e t a t i o n af quarks a s Bose p a r t i c l e s would a l s o lead t o the wrong r e s u l t t h a t the wave function would be s the i n t e r e h a n p o f two protm&. problem would be? remved.

I f quarks obey para-stetretics both of these

Me c o n t i n w our discussion of equation (39.7) now turning t o the region v

+

-, -q2 f i n i t e .

Here we expect 2W

which we have wrftten a s 2F?tW3

1

t o go a s v times a function of q

2

2Hv g(-q )I(-q

2

)

.

2

I n t h i s region c u r equation

(39.7) reads

a t f i r s t sight, for v

-+

=, we could forget -'-a

i n the 6 and obtain ZW1

9D

h(cr,o)da,

a czcnstant indepndent of q

2

asld evidently i n c o r r e c t ,

-

Elmever,

0 the "ceanstant" would, by (39.3) be f(o) which we know is i n f i n i t e , a s f o r small

% i s suggests t h a t we assume f o r smll 13 chat h ( o , ~ ) is singular.

ajx.

f(x)

X,

Sllbstitutfng i n the Antegral above gives

f o r small 8 ,

Thus

gives t h e -q g(-

2

) a s -q

2 2

dewndence of the t o t a l - v i r t u a l photon cross s e c t i o n , g e t s l a r g e approache

Finally we t u r n t o region 111: kept f i n i t e .

of MH -; '

There we expect 2MWl

only.

(~:-d)

v

+

-, large

-q

2

but

0, sgnB

(For v negative the s i g n reversea.)

-b

= sgn(*m)

Itence we have

s o i t is s t i l l , c o r r e c t ,

I92

Photon-Hdrsn Interactions

Wl/vp is a y m e t r i c s o we? c m s e t t h e range front Q t o l&u

'- ( v + i ~ ))

-

2 -h/( - v ' ~ + ( v + ~ c ))

.

with l/(v'-(v+ik))

.t

F r w t h i s equation (40.6a) follows.

For W which c o n v e r p e Easter we expect a corresponding uneubtracred 2

d%apersion r e l a t i o n

2

for q

W2 (q , v ' )

For qZ

Q these a r e of course t h e Kronig-Kramers

2

4

.

O

formula9 r e l a t i n g r e a l and

i m g i n a r y p a r t s of the index of r e f r a c t i o n ( f o w a r d l i g h t s c a t t e r i n g ) which

is necessary i f s i g n s l s a r e not t a corn out before they go i n t o a block of s c a t t e r i n g nratsrial,

We t r y now once again (msucc;essFully a s i t w i l l turn a u t ) t o t r y t o oBtktin s o w expected l i d t a t i m s on 2MWI b e c a w e we kaow i t is a caueaf (zero outside the l i g h t cone i n space) c m u t a t a r , v a l i d not only f o r l a r g e

V

Ttris time we s h a l l use behavior

For e x a q l e we h o w f o r t h e

but f o r my xeglon.

e l a s t i c s c a t t e r i n g , i f t h e proton were a point &arge we would have f o r v > 0, 2 2W1 Z H ~ 6 ( ( p + q ) ~ - d )= 2Hv6(q +ZHv). (To g e t the c o r r e c t s

-

p o s i t i v e and negative t h i s can be w r i t t a n = r e 2 6 ( q +2m?rv)

- sgn(v-M)S(q

2

-2Mv)).

2 Mv(sgnCv+M)

c o r r e c t l y a s 2Wl

This is o b d o u s l y cauaal

- as

W@

have seen

-

Ear i t coaes from perturbation theory o f fielcls. Now in the r e a l world t h i s is a d t i p l i e d by a f a c t o r ( t h e square of t h e 2 e l s e t i c form f a c t o r ) , a f a c t i o n of qZ, say f (q 1 which f e l l s o f f gradually. 2 f o r negative q , from q2

-

Q, to behave a s (_q2)-y f o r l a r g e -q

2

.

One would

expect such m d u l a t i a n of the f w c t i o n expected f a r point-like particles r o represent some kind of s m a r i a g of t h e point and t o perhaps irayrly a lack of causality

-a

lack which must be balanced by c o r r e c t contributions from t e w

o f f t h e e l a s t i c mass s h a l l (correeponding t o o t h e r resanances, e t c ) .

qlhue, c m

we n o t tnake so= requirelnents of behavfor i n o t h e r relZions, especially non-

2 2 e l a s t i c regions by our knowledge? of f ( q ) f o r negative q ?

trnfortunately n o t

(aa Cornwall, Corrigan and Mortm, (Phys, Rev, I)3 537 (1971)) show) i t i s possible t o s t a y m t h e e l a s t i c w s s h e l l fox negative q

2

and only a f t e r the

2 behavior f o r p o s i t i v e qZ and s t i l l arrange v i r t u a l l y any f f q ) which f a l l s o f f

Properties of Commutatorsin, Momentum Space Eafrly smoothly.

2

I n p a r t i c u l a r i f f ( q ) can be w r i t t e n i n the f o m

01

f o r negative qZ, i t can b e done.

W e can s e e why t h i s i s , and a l s o g e t a

b e t t e r physical f e e l f o r constructing causal functions by the following conI t is e m l e s t t o deal with the s c a t t e r i n g Emctioas T, fox these

siderations.

t o be carnal they must be t h e R u r i e r t r a n s f o m of a retarded c o m u t a t o r and thus zero outside a. fortsard l i g h t cone, its F.T.

Let a ( x , t ) he such a function and A(q)

and l e t b(x,t) be another such fmctlon,B(qj its F.T.

Thus the con-

volution of a(x, t ) and b ( x , ~ )is obviously, by g e o e t r y , such a function (zero outsirfe Eomard l i g h t c m e ) and hence its P,T.

o r s i m l y A(q)B(q) is again

s a t i s f a c t o r y (a clausal s c a t t e r i n g function). We s e e t h a t cc&inatioas by arultiplieation (and addition of causal s c a t t e r i n g

The s h p l e s t causal s c a t t e r i n g

fuaetians a r e causal s c a t t e r i n g functions, function is

lq2 -

m

2

+ ic

(40.82

sgn q4]-1

We can generalize t h i s (multiply by eiu*' vector u

i n space time) t o find f o r any four-

that

is a c a u s a l function,

((qe)2-g+ i c

Thus the e l a s t i c s c a t t e r i n g function f o r point p a r t i c l e s

sgn( +M)I-'

(40 .IQ)

evidentfg i a causal ( a s we11 a s what you g e t by p u t t i n g -q f o r q).

tJe can

multiply t h i s by an expression l i k e (40.81, we see iq2-m2+ic

i s causal.

sgn v]-'

f(q+pjZ-d + i r sga(+~>j-'

This i s t r u e f o r any

m"

12 and hence arty s u p e r p o s i t i a a with v e i g h t

p ( u ) d p is, s o a s c a t t e r i n g amplitude l i k e (note ( q - ) ' d

is, by i t s e l f causal.

To g e t t h e c o r r e c t s

try for

= q2+2~\3

V

one need only add the

corresponding expression with u replacetd by -v; we w i l l suppose It is always; d w e awl n o t w r i t e i t : e x p l i c i t l y .

[you olight f i a d i t physically more s a t i s f a c t o r y and e a s i e r t o h t e r p r e t i f the f a c t o r is considered ss a form f a c t o r due t o v i r t u a l p a r t i c l e s l i k e a t each vertex,

p's

"Coae a t one v e r t e x would contribute a f a c t o r

CD.

&ere

id

is t h e mass squared of t h e v i r t u a l p a r t i c l e and v znea-slurerr t h e weight

of i t s contribution; g i s causal, (g(:]12,

We would then exlpect t o znultiply (40.10) by

one f a c t o r f o r each coupling; t h e r e s u l t would s t i l l be causal,

and possibly physically e a s i e r t o understand.) To g e t ul/v ( c a m a t a t o r ) from (40.11) we need only take i t s inraginary parC

In t h e region q2

G

O t h e l a s t term disappears and we a r e j u s t l e f t with

the e l a s t i c point charge s c a t t e r i n g a d t i p l i e d by a factor: f (g 2 ) given by (40.7) I t is disappointing t h a t t h e r e s t r i c t i o n s of c a u s a l i t y do a o t a f f e c t our

region of exper%aental observation p r a c t i c e t o g e t p(u)

even i f f a i r l y exact, knowledge of P(&

2

((1

< 02, F u r t h e m r e , i t w i l l be hard i n

exactly from Lnwledge o f t h e i n t e g r a l (-q2=)'Q

It Is n o t easy t o reverse the i n t e g r a l I n t o accurate

mless physical axgumnts (P dornlnance e t c .) e r e a v a i l a b l e .

Thus again we a r e t h r a m back i n t h i g p r o b l m t o understand the pracess physically ; the mthentatical properties da not help a s much a@we had hoped. 2

(We know t h a t i f f (9 ) f a l l s f a s t e r than l/$, nay a s (l/q214; then we

1

can conclude, by (4Q.14), t h a t

p(p)du

0. Such a r e l a t i o n is c a l l e d a

2 2 superconvergence r e l a t i o n -£(Q ) converges f o r l a r g e Q more rapidly Chan

.

199 .

Properriw af;Commutators in Momentum S p c e ttre f o m would suggest.

Again s l n e e f f a l l s ae (1/Q 2 ) 4 we can concjlude that.

wnp(w)dv vanish f o r n 2 2 2 expressed a s f g t q 1) with g(q 1 =

1

0, 1, 2 ,

kbternativsly, f can be

Lecture 4 1 The i d e s dfseussed a t the end of the previous l e c t u r e is i m e d i a t e l y genelcalizrible t a s c a t t e r i n g through an f a t e r ~ l e d i a t eresonance, say of mass Iv&2 , = h.

c a l l E?M-:

A point coupling would give s c a t t e r i n g a s (q 2+2Mv -h

We can m d t i p l g by m y form f a c t o r , say with a p(h,w).

+ ic

sgn(v+M)) -l.

We; are thus representing

things by a e m of a charnel rczsonances each of which has a square f o m f a c t o r

Tlre t o t a l s c a t t e r i n g fronn a l l t h i s

(S

c b a e 1 resanmce representation)

would t-hen be

The W2 which goes with t h i s (the i m g i n a r y p a r t of ?/v)

For q

2

c

0 O t h e f a s t t e r n vanishes end v: have f o r v

0 a suparpcaitlon of contr%butlons f a r each e f f e c t i v e

Bave we not gone i n a c q l e t e c i r e l e ?

3

is

O slaply

with a f o m f a c t o r

Ckzr o r i g i n a l expression giving

196

Photon-Hadron Interactions

(except f o r photon p o l a r i z a t i o n f a c t o r s ) W was of t h e f o w l

This looks j u s t l i k e (41.4).

2 The 6 is of course 6(q +~Mv-X) s o we i n t e r p r e t

f(i, -q2) a s

S

Z~G~/J(~)/X>~~ d over s t a t e s of

B

given massZ =

$+

A.

(At f i r s t t h i s would seem t o meke a function of Q ~ ,t h e space p a r t of the momentwn t r a n s f e r squared, i n s t e a d of qZ = v2 -9 2 ; but i t is t h e same because the 6 f m c t i o n r e l a t e s v t o 'Q

and i t can be e q r e s a e d e i t h e r way.)

63e s e e

2

f ( h , -q ) must be p o s i t i v e f o r -q2 > 0 ; of course, s i n c e the lowest e l a a r i c

Mx = EI (X = O) is separated from the eontinuutn a t Mx =

2 (hth. = 2 % M n ) ,

The function f (h, q2) and hence p (A, p) w i l l have a 6 (A)

cont riburion and a f t e r

t h a t the i n t e g r a l i n (41.3) w i l l s t a r t a t an I n e l a s t i c threshold hth t o *, We m y noE have gone i n a coerplete c t r c l e ,

F i r s t we now know (a) t h a t the

2

weight f a c t o r f ( h, -4 ) mst be e x p r e s s i b l e i n t h e form ((31.1) a d (b) we know what t h e f m c t i o n looks l i k e i n t h e experimentally unavailable region +q2> O (see 41.3) and, of course, t h e s c a t t e r i n g Emetion

(41.2) t h a t goes with i t ,

But do we know t h i s ?

IJe only proved t h a t the f o m (111.2) was causal; we have n o t proven t h a t every c a r n a l function must be expressfile as (41.2) and a t the m m n t we do n o t Chink we can. Since (41.2) i s causal i t mst be expressible i n t h e BGS form (39.6).

Qne

way t o do t h i s (suggested by Cornwall, Corrigan and Norton, Phys. Rev. I)3 536 (19711) is t o co&ine dens&aatora ( f t is e a s i e r t o use the Fegnma amplitudes replacing

t o gat a s i n g l e denorainator.

Thus we e a l 1 a =h@+y(1-8) m d i n t e g r a t e on a by

p a r t s t o prove 00

00

Of course t h i s can now be simplified.

I f frotn b(o,B2 we could always fSnd a

Properties of Cornmufatorsin Momentirr?~ Space r>(h,p) which would give t h i s h we would have a proof ( a s s d n g the DES repres e n t a t i o n is proved) t h a t (41.2) is a l s o a necessary f o m . cmnot be done a d

We suspect i t

i r i s poBBible t h a t (41,2), although very physical, is not

the complete expression; but o t h e r f o r m (other types of d i a g r a m o t h e r than e channels) &@fithave t o be added t o sentation,

it

- o r "abaonels"

t o g e t a c e w l e t e repre-

This is a good problem.

W for a l l q

2

i n t e r n of W f o r q

2

61

We have a f o m t h a t s u g p s t s t h e qaestion a s t o whether knowledge of the f a c t t h a t W is causal, and knowledge of

r. t o f i n d i t f a r a l l q2, v.

irs value f o r negative g' only, enables

We a r e nor now concerned with t h e p r a c t i c a l f a c t

2

t h a t knowledge; of f ( h , -q ) t o a ggven l i m i t e d e x p e r i m n t a l accuracy does n o t p e d t discovering p (?,,p) 2

f(A, -q ) f o r q2 > 0.

very well. and does n o t m t h e m t i c a l l y alone d e f i n e

Rather we suppose W

2 p e r f e c t l y koom f o r q < O and ask

f

t o what e x t e n t T is defined e v e r M e r c l , This has g r e a t i n t e r e s t f o r there a r e q u a n t i t i e s , such a s e l e c t r o ~ ~ l a g n e t i e s e l f energies, which can be defined I n t e r n o f T as i n t e g r a l s perhaps (e.g. 2 4 T(q ) d q/q2).

I f T were uniquely determined by W i n the ertperimentally

a c c e s s i b l e region we might search f o r expressions f o r these I n t e g r a l s d i r e c t l y i n t e r n of t h i s W a t q Given W(q

2

,V)

2

Q.

(Gottinghsarn farraula f o r s e l f e n e r ~ , )

2 i n the a v a i l a b l e region f-moraftntuar l i k e q 1 how m i q u e i s

2 W(q ,v) i n t h e energy-like region?

2 2 let Wa(q ,v) and Wb (q ,v) be two c a u s a l

2 functions i d e n t i c a l i n t h e q .r Q region,

t h e i r difference W

Wa

What form must i t have?

- Wb.

Let

2

Wd(q ,v)

m

study what is p o s s i b k f o r

O f o r q Z < O and hence is causal.

We s e e i m d i a t e l y

sgnv6(q2-m2) is such a function.

W

t r need nor be zero because

To g e t the most general f o m we use wnon's

theorem rJkridir says Caince i t is c a w a l ) t h a t i t mwt be e x p r e s s i b l e fn t h e fom

where @ is non-zero i n s i d e t h e region where the hyperboloid ( q - u ~ ~ mdoes s ~ not p e n e t r a t e t h e region S of q space where we know Wd vimishes.

S Le t h e

e n t i r e q2 < O region.

It is seen that every hyperbolofd cute S unless

3 % ~the m s t general Earn f o r

-

U

0.

% is

er,

*ere:

by definition

#(X)

= O f o r x < O,

Thus complete bowledge of W ~ ( ~ ' , Vi n) the experimentally available 2

q c O region p e m i t s defini tion of kfl e v e w h e r e within an a r b i t r a r y constant, 2 2 (indtependeat o f v) f m c t i o n of q f o r positive q , T f a d e t e d n e d a l s o up

t o the function

S t r i c t l y t h i s s x g w n t is not valid, gradients of S fmctfons i n space time could be vled i n @son's theorem.

Hore correctly, a t each positive qZ the

f m c t i o n W is d e t e M n e d bp the bebavior of W{q 2 ,v) f o r q2

4

O up t o an mknom

2 p o l p o d r a l i n v, the coefficients of which a r e a r b i t r a r y functions of q

.

Physical argmenta about large v as~raptoticbehavior would have t o be wed t o li&t the degree of these p o l p o a a l s .

The scaling lidt f o r

t of the farm vQl{q f a c t that W is odd t e l l s us that (41.7) a u ~ be rurd that T1 is dete-aed

up t o a function of q

fmetiou is not zero f o r q2 by W.

4

2

0 and the

X

2

1 (for v

&vcm by (41.8).

7

0)

Note tt?ls

O so Tl i a not completely deternlned f o r q

2

O

This agrees with the dispersion r e s u l t (40,6a) where a subtraction had

to be made aad m a r b i t r a e f m e t i o a IT1@

2

,0)) l e f t u n b t e w n e d .

Electromagnetic Se

We now diacuss a flew places where knowledp of W o u l d h e l p ua caJeulare the e l e c t m d p a & c p r o p e r t i e s of protons and neutrons,

Since we have masured, i n W,

t h e eleetroanagnetic coupling of protons we nd.@t herpe t o use t h e a x p e r i m n t a l knmledge t o d e t e w n e the electromagnetic energy of proton and neutron and t h e i r w a s u r e d difference, t o comare t o e x p e r i m n t ,

Aa we s h a l l s e e t h e hope

is, a t present, f r u s t r a t e d because knmledge of W f o r q2

O where i t is avail-

able is not q u l t e enough t o d e t e d n e t h e e l e c t r o m g n e t i c coupling e v e r p h e r e (T)

- the a r b i t r a r y f w c t i o n of Slnce the

rfetemtned,

EmstJer

+(a) m a t t o n e d i n the prevliom l e c t u r e is not

f o r the p-n msa difference Le only one a m b e r we

a r e fruiirtrated w t l l we Can find a t h e o r e t i c a l o r experimmtel way t o deternine T uniquely

- f o r e ~ a l a p l et o

2 deternine Tl(q ,Q) of t h e dfsperslon r e l a t i o n (40.6a)

2 f o r q .: 8. Before we discuss t h i s by f o m l mathemtics l e t As La wefl k n m , the s e l f

micallg.

W

see what we can expect,

mss of a point spSn 112 p a r t i c l e diverges l a g a r i t h -

2 For the e l e c t r o n b(m ) = m2

Zn i n

J

where A i a some upper cut o f f

f o r e l e c t r a d y a a d c s i f we replace the photon propagator

This bra

2

A s i @ l a r i n f i n i t y f o r the AM

i a experilnentally undefinable,

m p a t i c f o r t h e proton xrould a l s o '""proton

- by i t s e l f

electro-

not be observilble, but

'""neutron

is, i n f a c t , observable and is nzeasured t o 5 s i g n i f i c a n t f i g u r e s .

calculate i t

2

- o r even

s e e what order of arapitude i t i s ?

-

Can we

f o r e x m p l e does

our present theory say i t must be i n f i n i t e ? As long a s t h e electromagnetic i n t e r a c t i o n of t h e nucleons were v s t e r i o u s

- but now t h a t we have so=

one could always sag anything could happen

knwledige

of them we must tursver m r e s p e c i f i c a l l y , The divergence occurs from high frequeaciers and a t f i r s t it was thought the hadrons d g h t be s o f t a t high frequency and t h e electrorrtagnetic s e l f energy convergent.

But now we know f o r i n e l a s t i c s c a t t e r i n g a t l e a a t they look l i k e they of p a i n t - l i k e c o m t i t u e n t s .

a r e nra&

energy must: dfvergel

Qf course, the proton could diverge and the neutron also-

only the difference need conwrge

Wlp-Win

Does t h i s point-like behavior mean t h e

- but

t h e d i f f e r e n c e i n point-like s t r u c t u r e i s

is a l s o f i n i t e and point-like i n the s c a l i n g l i m i t .

I t is t h e coincidence

of the &-function of t h e photon prepagator aad of the e l e c t r o n propagator which &es

t h e divergent s e l f energy

- and natir we s e e t h e protonI

t h e neutron, and

proton minus neutron a l l have s i n g u l a r behavkor an the l i g h t cone s o i t a t f i r s t looks l i k e dfvergenw i s inevAt&le, Let us estimate haw much,

Since we a r e d i ~ c a s ~ ~ ti hneg h i & energ?r b e h a d o r

we can use t h e i d e a t h a t protoampartons,

Clearly t h e s e l f e n e r e d i a g r m of

rnost i w o r t a n c e a t high energy i n the s c a l i n g lidt a r e when a photon i s e ~ t t e d 2 and absorbed by the same parton therefore, ss far a s t h e divergttnt (Rn h )

-

p a r t is concerned i t i s a s i f each perton g e t s a. s h i f t i n mss p r a p o r t i o n a l t o 2 rn?eiZ~nh /mi'

&mi2

where ei, mi a r e t h e charge and mass of a parton.

m c h does t h i s change t h e mss of t h e proton? acight t r y t o c a l c u l a t e t h e change i n E the changes i n e-p 2

'nuclean

2px

-C

2

blil partons x

We caslnot honestly awe One

- PZ = M2/2P by c a l c u l a t i n g the sum of

of each parton, -=a

Hw

We would f i n d

PP(,)gnh2dK

There a r e objections t o t h i s (by the way i t i e a l s o even m r e divergent since f

ea

l f x and the x i n t e g r a l cannot be c a r r i e d t o 0).

The energy is n o t

Electromagnetic Self Energy j u s t t h e sum a f the k i n e t i c energiers of t h e partana, interaction energies a w n & the partons a r e a l s o involved,

%is Is r e f l e c t e d i n t h e dangerous Eomula con-

t a i n i a g t h e mms squared of a partan 2A

up t o now.

- a thing we s a i d was meaningless t o

The d i e t r i b u t i o a of partclns I s r changed h f i r s t order a l s o s o

2

(Et d g h t be thou&t we a r e

t h e t o t a l ro change Is not c o r r e c t l y evaluated, r i g h t t o take wave funetion

$I

1 AV!$I>

- but

of the Lagrangian

AV

- in

fox t h e perturbation on t h e B d l t a n i a n w i n g t h e o l d 2

hro a*a i s not t h e change f n the Wamiltonian, only

the Haaltoniitn there are

o t h e r e f f e c t s on the

i n t e r a c t i o n t e r n throu&h l/6Eactsrs e t c . , s o we? have nob computed t h e e f f e c t of t h e perturbation c o r r e c t l y ,) It may well be t h a t mfZ

0, o r is e f f e c t i v e l y zero (a suggestion t o me

due t o Zachariasen) due t o i n e r a c t i o n s (or ae a nnatter of p r i n c i p l e ) s o t h e 2 logarithmically divergent perturbations bi " l p n d l n g on RnA n e w r a r i s e , m y be so, but we do not know,

Thie

We mst t u r n t o w r e d e t a i l e d qurtntitativt?

analyaesl i f we a r e t o t r y t o study ?&is

further.

E l e c t r o w g n e t i s mm8 s h l f t s come from t h e emission a d r e i b b s ~ r p t iof ~~

qED f e l l us i t is of t h e photon.

P / f JP (2) J U ( l ) lT fp>6+(s21Z)d~ where 6+(s212) is t h e propagator

Hence by trzkiag Fouxier t r m s f a r t n

Me, have w r i t t e n

aa t h a t

T M LJ

42 ( T ~ +([l ~ Z / q Z l ~ 2 w ~ 1 1 / 4 )

The expression i n square brackets has the imaginary p a r t (l-v 2 /q2 )W2-Wl which i s t h e contribution froon l o n g i t u d i a a l photoas,

Zf partons a r e s p i n 1/22 i t

f a l l s f a s t e r with v thao does Tl whose imsginary p a r t W1 s i n p l y s c a l e s t o f ( x )

i n t h e Bjorken l i m i t .

Ue w i l l hereaf t e r just write T f o r T1+l (1-v 2 /q2) T -T ) / 4 2 l

and suppose the imaginary p a r t W

2 2 Wl+[l-v f q )W2-W4/6 s c a l e s t o f(x) i n the

Bjorken l i M t .

h c t u r e 43

h we have seen we can write f o r the e l e c t r o a p e t i c w s e e f f e c t

e r e OT

m

TUll

.

We must i n t e g r a t e t h i s over a l l q

2

but we have seen T is

determined by i t s behavior f o r qZ ' O (where experiment can say eoaething about i t ) sa i t is possible that mybe (43.1) can be written I n t e r n of T for q2

O only. How t h i s can i n f a c t be d m e

was shown by CottIngham.

Be showed

that i n the four-dimeasianal i n t e g r a l the contour on v could be &=gad passing any s i n g u l a r i t i e s ) Erm the r e a l l i n e v axis v

iw, w =

-oo

to

we can then write d4q

Now replace

-(U

2

2

-oo

to

(hshaw how i t Is done l a t e r ,)

a?,

dw2n9d~'

) by {-q

2

1,

,

q2

v2q2

m

(without

t o the imginary Suppose i t is t r u e ,

-u2-Q2

Q =

t o get

(Cot tin&=

formula)

Nw the quantify is a l l right f o r -qZb u t is completely unphysfcal. f o r v is

laaginary.

Me can define i t however by a n a l y t i c continuation by our disparsfon

r e l a t i o n {SO,ba), settXng

V

iw

We can carry out the i n t e g r a l s on u t o get

This then succeeds i n g e t t i n g an e x p l i c i t f o m u l a f o r the s e l f energy in f e r n of wtq2,v) i n a region accessible t a experiment. 2

by the unknam term T(q ,O)

However, we a r e f r u s t r a t e d

.

2 Z t i s e s s e n t i a l t o kn&w somthing about T(q (0) if we a r e t o be a b l e even t o deternine ðer the s e l f energy divergee.

Let us look f i r s t a t the c s n t r i -

bution f r m t h e eecand t e r n i n ((43.5) i n the e c a l i ~ gregion fraxn whlch divergences could c o m e Put -q2 = ZMvx and consider W a s a fvnetion of x and v, W(..") Tor l a r g e v cmverges t o the lMt f(x),

dx vdv

-1

- 1;

vhieh

"&e t e r n becotlles

utx,v>

o r f o r l a r g e v, where t h e square bracket f s -(2&/v)

2

14 we g e t

xf ( X ) dx, the

The v i n t e g r a l diverges logsritharically with coefficient

f r a e t ion

of trrcmentma c a r r i e d by the charged partons each wei@ed by the b a r g e squared, 2 2 Of course a cut-off of electromagnetkm is used i n (43,5), d ( q )/(+I ) is replaced by This provides a cut-off f o r our v i n t e g r a l (of order f / 2 & ) diverging as ko A

2

so the parr

f

has a eoef f i c i e n t zf(X) h. 2

However, t h e other t e r . in T(q $ 0 ) could a l s o produce a fin A' term f f i t only f a l l s aa C/(+ 2 ) a s -q2 -, G*,

2

(Zf T(q ,O)

cei(s)ds/(q

2

44

2

1 this C

1,

Z t is therefore possible t h a t these divergences from 1: ernd W cancel and t h a t the

s e l f energy (or a t l e a ~ tthe protm-neuf ron dLf ference) is f i a i t e and calculable, m e r e a r e two viesrs we c o d d take a t present,

We know of course the p*

=ss difference converges so Let us t a l k about T snd W f o r the p-n dlfferenca a t least.

In p r i n c i p l e T could be determined bp experfirnent and is therefore defined

phyaicall*y, e i t h e r of the f ollowing could happen. (a)

Equation (43.5) with t h i s T s t i l l ggves a logarithmic divergence, The re-ason is t h a t our theories a r e wrong f o r high energy; t h i s and the e l e c t r o m g n e t i c s e l f m e r g y calculatLon of QED a r e both wrong and w l l l both be fixed by t h e s a m modification of r e l a t i v i s t i c quantum mechanics a t high energ?t e o m day s o be found,

Cb)

The T is such t h a t the i n t e g r a l s converge and agree with t h e experim n t a l mas8 d i f f e r e n c e s

(c)

2

The TCq ,O) ( f o r q

2

G

.

0) is i n f a c t n o t p r e c i s e l y definable experimentally,

t h a t i t Is t o soroe e x t e n t a r b i t r a r y

- hence

that, our theory is not a b l e

t o c a l c u l a t e t h i s =ss d i f f e r e n c e p r e c i s e l y and rnust be " r e n o m 1 i z e d e t ' 1 believe i n t h i s case, i f partons a r e quarks only one r e n o m l i z a t i o n

constant, corresponding t a the e l e c t r o m ~ e t i c=ss

difference of u

and d quarks, would s u f f i c e t o make a l l the hadron s e l f e n e r m d i f f e r e n c e s converge simultaneously, As Zachariasen has suggested t o rae, t h e b e s t (.roost l i m i t i n g ) t h i n g t o

do today IS t o aseum (b] is t r u e , and may have p r e d i c t i v e value, l e a r n t h a t (a) must be s s .

This puts r e s t r i c t i o n s on possible t h e o r i e s

Xf i t leads t o a paradox o r inconsistency we

Zachariasen has shown t h a t a l l w i l l be convergent

i f t h e equal time c o w u t a t o r o f J and =del

3 vanishes

[J

P'

iy] 0. I n

the q u a r t

i t c o r ~ s g o a d st o quarks having zero r e s t =ss.

Z believe, t h i s Is a very good problem t o work an.

I myself haven"

found

enough t i e while preparing theae notes t o analyze i t i n a more elenensary o r c l e a r fashion f o r you,

2 Now can we ever hope t o g e t a t T(q ,0) experimentally o r t h e o r e t i c a l l y ?

Xt is t h e forward s c a t t e r i n g amplitude on a proton of a v i r t u a l photon of 2 mass -q

.

It would be iavolved i n a wo*lectron

forward s c a t t e r i n g e+e+p

-+

e++p v i a t h e dlagram

R i s i s not aa expergmnt t h a t can be done,

2 But knwledge of T(q ,v) a n p h e r e

EIecrromagnetie Self Etiergr would be of a s s i s t a n c e because the dispersioa r e l a t i o n s can be used t o convert i t t o knowledge of T(q

2

,Q); s t i f f no experiment suggests i t s e l f .

It is i n t e r e s t i n g t h a t T(O,O)

can be obtained t h e o r e t i c a l l y

Coapton s c a t t e r i n g from a proton ofIP a r e a l (on s h e l l ) photon,

- the forward

For Q

-+

0, v

-+

O

we have vary long wave length slow f i e l d s to whsctr, of course, the proton looks t o be simply a raassivr: point charge of =ss

M,

It s c a t t e r s then as i t would

clczssicslly [or man-rela t l v i s t l c a l l y v i a Schrddinger equation from t h e

Q2 +

-b

AeG

tern) thus (called Ray1eigh s c a t t e r i n g ) It glves 2 T(O,O) m

- %.

NOTE: Wow to r o t a t e contour to get Gottingkara f o m l a ; Use WS t e p r e s e n t a t i a n f o r T

Call E

P

mid note t h e slingufarities a r e a t v

F

Since E > BM, t h e palea below the a x i s a r e a l l f o r v > O,

;

Q - i ~ , +ic

llhe contour goes l i k e

the dotted l i n e , i t can evidently be r o t a t e d t o the i m g i n a r y axis.

Lecture 44

W e d i & k succeed i n representing AM

2

2 i n t e r m of W(q ,Q) f o r negative

2 q only, wilthout a t the s a w tim invalving ourselvee with another w h a m 2 f m c t i m T(q ,Q); and f u r t h e r each of two p a r t s i a i n f i n i t e and i t Is hard t o guess a t the difference, 2

Perbpps we should abandon t h i s and take 2

8

l a s t lnok 2

at: jwt the bI4 expressed i n t e m of W ( q ,v) for posirive and rmegaeive q

is given by

.

Zt

206

Photon-Hadron Interactions

(This is obtained by expressing

and i n s e r t i n g i n t o (42, l ) , W (v

i n the form

$6) depends

only on Q, the =pitude

of

4; and

To study the possible divergence of tf-re i n t e g r a l a t l e a s t , we go t o large 2 v and i n f a c t t o the s c a l i n g region -g 12Mv

x , hence Q =

v+m. We can write

is kinematically I.

2MM approaches the

; .

(consider 2W a function of x and v, ZW(x,v))

For l a r g e v the upper

lintit

for

X

fuaetfon f fx) f o r poeitlve x I n the scaling l i m i t ,

Far negative

i c approaches a s we have seen -f(-X); note, hotsever,that

-X

X

2 (positive q )

is not k i n e m t i e a l l y

DyndcalZy -x i s l i b t e t l i , W(x,v) e d s t a f o r x < -l but f o r large v f a l l s rapidly u n t i l there is nothing l e f t of order one. The eontributian of the scafing region gives But t h L s n l y says t h a t a dlmrgence h i & e r than l a g a r i t h a c vanishes, a Uling we expect a n p a y , Wx/Zv leading t o

Expanding the faetor ( /xf(x)(dv/v)&;

)/(2v+m)

s e e m t o give a tarn

t h i e i s not a l l , we s h a l l have t o Lnw to

2 order l / v haw M d i f f e r s from i t s Bjorken f i a t f o r bath positive and negative q ,

This suw up the probleni; experimat c m i n principle kelp m with the approach t o the limit f o r q

2

negative.

But we s h a l l have t o r e l y on theory t a

obtain the cmtribut?tans t o order I l v f o r positive q

2

before we can decide

whether d g diverges according t o present theories.

Waving f a i l e d wfth ftnrdamntal theory t o get i a f o m t i o n on etctrornagnetic

mass differences, we now turn t o nnrch cruder pictures t o diacms the possfble relatSons of 4 2 i n d i f f e r e n t terns ctf an $Uj o r $U6 multiplef. the laaguage of the quark made1 althou*

We do i t i n

mny af the r e s u l t s come from weaker

~eunuptiana, Iflire s i q l e SU3 sec. The proton, f o r e x a q l e , i n the low energy qmrk model is made af three

Electromagnetic Self Energy quarks, two

U

quarks and a d quark with t o t a l s p i n l 12 with wave function (2 uud

]p>

- udu - duu)

(+S+) symetrieed

.

(44.3)

The electromagnetic s e l f energy can be thought of as being ntade of two parts: a)

The s e l f enerp;y of the i n d i v i d u a l quarks.

We suppose & i s

t o t h e change squared o f each, thua 4a : a: a f o r u: 4: 2

proton t h i s csntr-lbutes bM f 2 N to

P

S

ifs

proportional

respect%veIy.

To a

= 9a (we s h a l l n o m l i z e a l l mass squared changes 2

2P1 of t h e proton a s a s c a l e f o r laeasaring a , t h e true. change i n M i s than P

2M a,) P b)

An interaction energy bemeen p a i rs which we take as proportkonaf

t o t h e product o f charges. r e l a t i o n of the p a i r ,

The i n t e r a c t i o n mmt depend on the wtwl s p t n

'&M w r i t e

@(l*)

i f s p i n is p a r a l l e l and B(1-y)

if

s p i n %sa n t i p a r a l l e l ; t h i s i s B(1WP) where P i s t h e s p i n exchange o p e r a t o r , We a d t i p l y by -2 f o r ud o r us, 3.4 f o r uu and + I f o r dd, ds, s d , ss p a i r s ,

call t h i s factor x

il "

The electromagnetic s e l f

operator

cam t h e r e f o r e be w r i t t e n

Thus

X t i s easy t o get the e x p e c t a t i a n of t h i s o p e r a t o r fox e v e q s l a t e .

on t h e proton t h e o p e r a t o r Bx alone does n o t s e e t h e s p i n s and gives (2(4-2-2)uud (4-2-2)udu

- (4-2-2)duu)+l.+/&

which is zero,

For the neutron, change a

t h e c o e f f i c i m t is -3 s o t h e r e is a c o n t r i b u t i o n

- 3f3,

Next we study 6yxP

therefore xf"(2uud

- udu - d u u ) + f + / g =

=1/8.t.2)uud

+ (-4-4+2)udu

+

(-4+2-4)duu)++C/&

and

Adding up the v a r i o u s c o n r r i b u t i a n s w e have f o r the proton

AM'/ZM

P

-

9a

+ 68y

U

t o d,

-

3.

We c m uldlculate the E=M s e l f naasses for evesJl p a r t i c l e of the l12

octet:

a s was done i n the previous l e c t u r e f a r the proton, the r e s u l t is:

Hence hM

We have three constants f o r four ms8 dJbfferances s o we have Ehe relation (m SSU3 r e l a t i o n ) (E+-c-)

- (p41 -

(We choose t o use P$

w h i e h f i t s well.

f o r no extremely good reason, the r e l a t i o n

AM but there l a l i t t l e t o chooole,it is -Snsi& the ewerimanta%

f f t s b e t t e r with

e r r o r f o r dk?

B@- 3 -)i. a

&Q.)

The values of the constants t h a t we get are 3a

-2.0

@(l*)

.24

B(1-r) = f .32

B

m

-78 NeV

MeV %V

p a r a l l e l spins

M V

a n t i p a r a l l e l @pins y

= -.69 HeV

The s i g n of tt 1s the oppczeice of h a t you would expect but then yort expect

+

and

it

m e t be r e n o m l i z e d , possibly t o a newtivcc value,

It man@generally

209

Electromagnetic Self Energy p a r t i c l e s wlth fewer u quarks a r e heavier, i . e . m r e p o s i t i v e l y eharged baryons are l i e t a r .

The s i g n of the 8 term from e l e e t r o s t a t l c repulsion is p o s i t i v e

as expected.

We f i n d a t t r a c t i o n of p a r a l l e l raagneto i n m s s t a t e but repulsion

of a n t i p a r a l l e l (the magnets a r e

OR

t a p o f each o t h e r ) which is c o r r e c t ; the

n e t repulsion i n the p a r a l l e l case is l e a s . GontinuFng i n t h e 56 $U6 raultiplet t o Che d e e i m t and supposing the constants a r e t h e s a w we can p r e d i c t everything:

nn2/2n (predicted) P

ss:a if-

110

G-

=2-

m

Sa

- 313(29)

- 4.72 - 1.28

an-- 3a + 3B(l+)

Tfie expexiunentarl d a t a is not good but there is a s e r i o u s discrepancy with a very recent experiment on b0

-

A

~ except ; f o r t h i s t h e s i g n s and

general order of s i z e s i s good, For pseudoscalar tnesons, noting t h a t t h e s i g n of the charges is reversad f a r i m t i p a r r i c l e s , and t h a t only a n t t s m e t r l e s p i n c o a t r i b u t e s we g e t ( c a l l B ' = @(l-y))

W e have two constants t o f i r with two p a r a m t e r a and have no predictilon,

We g e t B'

* l 4 , 3a

-2.38 again e a n f i a r z g t h a t a is negative.

close t o its value f o r t h e baryons r*rhich is what =R%

Xn f a c t a is

a e l a t i v f s t i c quark rmdel

t~-auldexpect, m g n i tudet

F u r t h e m r e the 8'

.

F i n a l l y we do t h e vector m s o n s ,

Here we need g(1-y)

*K The only d a t a on AM is t h a t f o r ' 2

is n e a r l y of the s a w order of

@(l-y)

- K*'

b f o r p a r a l l e l spina

which i a -5.7 2 1.7 HeV o r

-5.1 2 1.5 = 3a + 3b. X£ 3a = -2.38 t h i s gives B = -,9 2 5, which is P very bad, i t is of the wrong sign. X f t h e values f a r t h e o c t e t baryons a r e f2M

2

w e d we p r e d i c t bm /2M 2 1.91 P h t h i g system we a l s o have two o t h e r e f f e c t s (discussed i n Lecture 15)

a) t h e e l e c t r o m g n e t i c rnixing rnatrix between p' 3a

- 3b12,

aad wo, with off dlagonal t e r n asld wo which we fotlnd t o be

b) The a n n i h i l a t i o n term between

'fhe o f f diagoaal elernent 6 i n t h e Htasa matrix

is deterninczd by

p ,w i n t e r f e r e n c e t. be -3.7

m n i h i l s t i o n t e r n + , 5 l gives -4.2 2

energy. Sa

bm /2M

-2 &V

P

+

.9 &V.,

then corresponds t o 3e

+

.9 MeV.

Subtracting t h e

f o r t h e contribution of the s e l f

- 3b/2

t h a t b is i n f a c t p o e l t i v e and near

3.7 2 .7, 4-

% i s suggests i f

1.1 2 .4 MeV, not too con-

s 2 8 t e a t with the baryon va2ue of ' 2 4 f o r p a r a l l e l spins. rise:

This i s generally w s u c c e s s f u l .

SU6 does n o t , predsctgng b0 the *sons

- A*

things a r e very poor,

(comnpared t o 1.3 f o r baryons)

.

+

For thie baryons SU3 works but For 0.3 f o r t h e experimental 2.9 2.9.

Far pseudoscalars we g e t 3a

.X4

For t h e vector mesons the s i t u a t i o n is czonfur~td.

2

If we use mass d i f f e r e n c e s inatead of dn~/2M

P

c o n a t m t s c a m our 3a

-2.4, 8"

--1.9t @(l+)

= 6.20,

f o r the r u l e s , t h e baryon

t((1-y) = 0.98.

The predicted

211

Eleerromagnerie Self Energy .l0

- tit+ i n 0.8 s o only l i t t l e is gained here.

a l t e r e d (beeawe of t h e snrall s c a l a r s 3a = 7.0,

8'

1.0.

n

But the meson situation is

The constants c o w o u t f o r the peeudo-

mass),

For the vector m s o n s we have 3a

-4.8,

-0.3.

b

for

The B ( 1 9 ) fox baryans m y d i f f e r from b f o r vector nresoas and baxyonhj nay a l s o d i f f e r from B y o r pseu&oecalar =sons because t h e s i z e af the wave function is s o d i f f e r e n t s o the man I / r d i f f e r s .

The e l e c t r i c and

mgncttle i n t e r a c t i o n need rtot change i n the same r a t i o so i t is poseibla t h a t

b is negative f o r nreeons and p o s i t f v e f i r baryons, but i t is d i f f i c u l t t o s e e 2 why 8 s o d i f f e r e n r a s the Am gives. Wfig the v a l w of 3a should d i f f e r i n one c a e from t h e o t h e r i s n o t

2 Were It n o t f o r experiment X would deci& 3a c a l c u l a t e d f a r bar /2M is

clear, the s

P

e f o r pseudoscalar and pseudovector *ems

m d is 0.6 of t h a t f o r

9111s is because we a l g h t guess t h a t mass squared s h i f t @ due t o

baryonrrr,

s t r m g m e s a come

from a w s change i n t h e s quark.

2 The d i f f e r e n c e lin na

f o r t h e baryons i s about .40, f a r the mm088 about .2S o r 0.6 aa such, t h e self energy c o r r e c t i o n e f f e c t

- if

Thus

i t i s t o be associated a i q l y with a

proper mass chmge of t h e u quark w i l l a h o be 0.6 a s e f f e c t i v e i n IBesons a s i n baryons

.

Evidently t h e naive theory doea not work well, we do not understartd things s o well.

A m r e d e t a i l e d d p a d c a l theory 19 necessary.

case, mast p a r t i c u l a r l y the l a r g e bo AI

+

- P*

But i n m y

is most disquieting.

2 mss d i f f e r e n c e s There 58 one obsarvation ehar c m be made h e r e which suggest8 t h a t d p m t c

calculation8 rnight be posslble f o r s o m caabinatiolorus,

Hotice t h a t a is the

s e l f energy t a r n , pcrssibly involving h i & fireqmncy behavior, but f311Cy) is due to m t u a l

i n t e r a c t i o n and ought t o show no diverg;enccz?. Certain co&inatfons

of w s 8qr;tared d.iffc?rences do n o t iavolve a .

Arl these involve t h e b I energy.

They a r e

2 i s o s p i n p a r t of t h e electroaragnetic & e l f

This energy depends on t h e products of two current operators JJ each

c f which. contain8 61

O o r l, and can w k e t e r m Khose p a r t s a r e

dX

0, I o r 2 ,

-

1 I(I4-1) respectively. and IZ2 5

I n terms of Iz these go a s constant, Tz. The constant (AI

-

the A I T'

- L-

0) p a r t i s e q e r i m n t d l y l o s t i n the s t r o n g i n t e r a c t i o n s ,

1 o r Iz t e r n . a r e measured by differences proportional t o Iz, l i k e

o r p-n.

But the bI

2 t e r n e f f e c t is proportional t o :I

measured by the differences (I* ' C We a r e calculating the

&IZ

-

+ C-

- 2z0) mantioned above,

O coarponent of t h e A I

but t o i s o l a t e i t think of c a l c u l a t i n g an a r t i f i c i a l P I z i f electromagnetic current J were AIz

and is

-

+ l (instead of A I z

2 e f f e c t , of course, +2 U a t would a r i s e

1. UP

Q), A I

mm t then think o f c a l c u l a t i n g an e f f e c t of two currents l i k e J+ J-1.

.

Now we s e e why a does not a r i s e , and why i n general, &oesib&y the i n t e g r a l @

i n calculating t h i s might converge rapidly a t high v, provllded parcons a r e quarks.

For i f all, t h e current c a r v i n g fwd-ntal f

isospin 112 i t Is i w o s s i b l e f o r two J succession a t high enc;r@, contribute only t o A I

operators (partons) have

t o operate on the @amet parton i n

%us t h e v i r t u a l photon exchange d i a g r a m l i k e A

1 and a r e I q o s s i b l e f o r bZ

p

2 , only B a r e a l l w e d f o r tar rong interaction photon

.-.-.-

Baryon

parton

Large v i r t u a l lnoeaenta of the phbton is posegble i n A no m t t e r how s o f t the s t r o n g i n t e r a c t i o n s a r e

- if

the i n i t i a l rrrrmntw distribution o f t h e

partoncl involves only slow ones so can the final, s t a t e .

But i n B, i f the

virtual, photon wmntum is very h i & i t is unlikely t h a t t h e s o f t s t m n g int;ersction can p u l l the partons back together t o give rnuch o f a diagonal m p l i t u d e tlo be i n the wdieturbed baryan atate. again.

As we found. i n the previous l e c t u r e , d p m i c c s l c d a t i o n s of bT

a

2 ass

differences should be f e a s i b l e by s we know t h e o r e t i c a l l y and e x p e r i m n t a l l y &out the expected behavior of the

Electromagnetic: Self Ettergy Of p a r t i c u l a r i n t e r e s t would be a study of the

necessary m t r i x elements, 'lt

213

- no .atass difference, .c.

The e l a s t i c . t e r n t o n t r i b u t e s alarost a l l of t h e n

as a s i q l e c a l c u l a t i o n w i l l show.

- no mass

difference

The d i a g x w a r e

p/s

P, v

71

The pion form f a e t o r is d o d n a t e d by the f a c t o r on P

2

p

reaonrtnce, we t h e r e f o r e F n c l d e a

2 2 / < q -m ) a t each photon coupling s o fke -ss It

( 2 ~ ~@P 3 ~ -q )1 p - 6

(P+ 2 To order zero i n mr /$2

2

2 - aann

difference is

d4q/ (2711

4 (46 ,l1

2

UU

9

the f i r s t term i n t h e square brackets is 1, t h i s gives

S The i n t e g r a l is r e a d i l y p e r f o m d r e m a k e r f a g t h a t fn four diraensions d q

a f t e r I n t e g r a t i n g over angles. (&m210

The r e s u l t of the zero*

order caltculatlon is

3e2m 2/4n, doing the next order gives P

h t e m of &m m n t a l 4.6 &V,

Ir

2 + Am /2on thSs i s n

- no

a:

4.1 &V

a s conrpared ts the experk-

E s t i m t e s of contributions of higher i n r e m e d i a t e s t a r e s

(&%eh could be done using FKR's

m d e l ) should be small.

A s i m t l a r r e s u l t is fovnd f o r E+

+ E-

- 2z0.

It seems as though t h e

e l a s t i c t e r n &ready gives a l m s t t h e complete r e s u l t .

2 2 2 n q dq

Bstimtez, of h i @ e r

resonmces give l e s s than 20%, and the r e s u l t is i n good agreeraent with t h e data, The contributfons of t h e e l m t i c term t o s e l f energy differences o f

172 l381

b a v o n s were calculated by Grose and Fagelis, Phys, h v . Sli

3

f o r the magnetic momants, and GE and C& varying l i k e ( 1

619681 using

+ q2/.71)-2.

They get bM(elastic2

P* C'

+0.79

-

C"

6'

- 6-

-0

.".-2

::

AM(exp. )

- 1.29

MeV

+ 0,16

- 3.06

- 0.88

- 4.86

- 1.10

- 6,5

&V

1 .7

( M s t of t h i s i s due t o the charge, s i a e e there a r e factors of q f r m i n the ntagnetic part.) to +1,8 e x p e r i w n t a l , tern.

Nate & a t atl

+ C 4 C-

- 2~'

4-1.54 &V

y 44y

Ir.

1.1

( e l a s t i c ) cornpared

we e x p c t e d this should be dominated by the e l w t i ~

(kcinrcirt i n t e m d i a t e s t a t e s give less than .l and pres

ones even l e a s ,l Note t h a t other c d & w t A ~ n s(other t h AE

-

2) do not involve "a@'

(see l e c t u r e s 44 and 45, "a" i s the EM s e l f energy of a quark) such as (p-n)

- ( 9 '-3 -1,

In t h i s case one carnot skaw t h a t these dif fereacer, a l s o h v o l v e

a product of currents su& t h a t each current does not a c t on the same quark, C suggesta t h a t we may be able t o g e t a t But the f a c t t h a t "a" i s ~ R O involved

t h i s difference a l s o by estimating m t r i r e l e m n t e t o varioua know s t a t e s w i n g , f o r e x m l e , the quark ~tadel. Wwever, tkr? e l a e t i e term f a i l s u t t e r l y t h i s t i m Am(e1astfc)

+1.9, A M e q . = 4.2 2 .7.

look a t the contributions from the sealing region.

ay? To explain t h i s we Let u(x) , :(X),

s G), ;(X) be aa described i n l e c t u r e 31, then vW2/x

3

d(x)

f (X) f o r p, n, 2

, ;(X), and

iki

(Tkre neutron is obtalned from the proton by replacing u by d and vice versa, The 3'

i s l i k e the neutron but s replaces u and vice versa.

'E but u replaces d and vice versa,)

m e 3 - i s lib the

B e r e f o r e the scaling Emetion f o r (p-nl-

21.5

Electromagnetic Self Energy

- 2 ""1 is $ ( u +

(Z0

+s+a

h i & frequencies can corn i n .

- 2d - 22)

which is n o t n e c e s s a r i l y zero s o

I n the w d e l of valence quarks plus s e a i t is

zero but we do not b e l i e v e t h i s t o be likel_y, For f u r t h e r d e t a i l s on a l l these w t t e r s of EM s e l f energy s e e an a r t i c l e by W, B, Cotringham i n "H~adronic I n t e r a c t i o n s of Electrons and Pro tons ", C~umings and Osbom E d , Academic Press, N.V.

1971,

Mcturre 47

Ve now go on to discuss o t h e r e f f e c t s involving two photon couplings. The Cowton e f f e c t is t h e most c l o s e l y r e l a t e d t o what we have done,

I f the

s c a t t e r i n g is e x a c t l y i n the f a m a r d d i r e c t i o n the s c a t t e r i n g m p l i t u d e i a given by T

QV

2 (q ,v) f o r q2

thus T is the

spin

0. We previously meant the average over proton spina. averaged f o w a r d s c a t t e r i n g , we could a l s o measure f a r

s p e c i a l s p i n d i r e c t i o n s of t h e proton.

The imal5;inary p a r t o f tfte f o m a r d

s c a t t e r i n g is, of course, the t o t a l c r a s s s e c t i o n cr discussed before.

l i k e ((97

+

(E.g.

67/&jub,

a

YP

YP

or a

Yn

wtrich we have

showd resonances a t low v, a f a n o f f perhaps

and (97 4- 43/&/pb]

f o r neutrons.

The d i f f e r e n t i a l cross secegan can be f i t t e d with

For m e r g i e s from 2 t o 7 GeV we g e t around 6

16 GeY, A with A

IJ

-

8

&vd

7.4, B

P

For energies from 8 t o

is c l o s e r ; there i s some s i g n o f a quadratic term A t 2.0,

np a t 9 GeV has A

9, B

+

Et

2

(This is lnuch l i k e hadron d i f f r a c t i o n s c a t t e r i n g , @,g, 2.S,)

Thus photon d i f f r a c t i o n looks very much the

s a w a s w o d d be expected f o r hadrons except f o r t h e very lauch snraller c r o s s s e c t i o n , of course.

kle now discuss the forward s c a t t e r i n g i n mere d e t a i l including s p i n e f f e c t s . The f o m a r d amplitude w y be w r i t t e n

as a s p i n m t r i x operating between s p i n statess of t h e proton i n t h e lab. systeas,

The various laeasured q u m t i t i e s a r e expressed i n t e r n of El and E2 a s follows:

The t o t a l cross section is the imaginary p a r t of the diagonal ( i n spln) scattering

The foxward d i f f e r e f l t i a l cross section f o r =polarized

This is h-.

forward

scattering i s

The r e a l p a r t of f l c m be obtained f r m the imaginary p a r t by a dispersion r e l a t i o n , (Eq. (40.6a) f o r q2 = 0) where we use the f a c t t h a t fx(o)

2

-e

/M.

This has been evaluated (see Dammhek and GiXmn, Phye. Rev. X)1 l319 (l9705 o r Busckihorn e t a l ,

241 (11970) and the sw of the f i r s t two

Phye;, L e t t ,

t e r m of ( 4 7 . 3 ) is compared t o the expcrrimntal (do/dtIo t o see how b i g the l a s t t e r n is.

They agree within e r r o r s f o r v from 2 " 5 t o 17 GeY s o the

emtr2bution is l e e s than 10% over t h e e n t i r e range.

( B e contribution

g r e a t e a t from the f l r e t t e r n i n (47.35 above about 5 &V,

W

I f212 /v

l8

The second is 15%

a t 2 GeV and f a l l s away a t hip;her v ) ,

-

2

: p v where iiAis the 2M molazlow p a r t of the = m e t i e mrownt of the nucleon i n nuclear =ametens.

W e a l s o how, f o r small v, a s v

G =-

'=L

"1. where a

i

+

and B

-r

0, IZ(v) =

?l

(47.5)

91 II

a r e d i f f e r e n t i a l cross s e c t i o n s a t fixed t Ear in~oraing

photons polarized pergendlcular o r p a r a l l e l t o the plane of callisiczn, t = O

For

Z must be zero, of course; but within the l i d t s of experimental e r r o r

(210% f o r -t e .2, e20X up t o -t

' 6 ) i t is zera up t o t:

-0.6,

(The

average of C f a r t = .l t o . 7 is .Q2 ".Q6,)

Me have discussed the s i z e of da/dt f o r yp i n r e l a t i o n t o (VDM)pp cross section (see l e c t u r e 201, i t is twice l a r g e r than VDM expects,

The a s ~ e t r y

produces no problem f o r the correspandi-ng a s F e t r y I n t h e p c m e , i t 1s a l s o very snnall;.

'This is not w e ~ p e c t e d , s channel h e l t c i t y conservation a l a o

Electromagneiic Self Energy expects the s a r e s u l t ,

The question is:

With i n c i d e n t 1i&t i n t h e z d i r e c t l o n

with x p o b r i z a t i o n do m r e photons s c a t t e r a t a snzall angle

i n t h e d i r e c t i o n x o r i n the d i r e c t i o n y l

SQ

Ctraneve~se)/v

From t h e point of view of d i f f r a c t i o n ,

c u r r e n t s generated by t h e incident. wave must be adequate t o produce t h e c o r r e c t f o m a r d s c a t t e r e d wave t o i n t e r f e r e with t h e incident wave t o a c c o m t f o r t h e mese

l o s s of i n t e n s i t y o f t h i s wave represented by the t o t a l cross s e c t i o n .

c u r r e n t s a r e obviously l i a t e d t o the s p a t i a l region of t b e proton, and s o they produce s c a t t e r e d waves i n o t h e r d i r e c t i o n s , t h e w v a l eAt of d i f f r a c t i o n from the proton, jwt a s i n hadronic c o l l i s i o n s .

But these c u r r e n t s m a t make pure

x p o l a r i z a t i o n a t l e a s t i n t h e forward d i r e c t i o n , they a r e x d i r e c t e d c u r r e n t s . In o t h e r d i r e c t i o n s a t sm11 m g l e s we have the same i n t e n s i t y f o r d e f l e c t i o n except f o r a cos Blab, therefore E -t

m

(1

P

1

- cos 4ab) / ( l + cos

X

and y

- eZlab profection f o r x d e f l e c t i o n , 4ab,)

0 . 6 , v 2 3 . 5 where t h e d a t a is taken.

@iab-t/2v2 */2

;-

+

.03 f o r

m u s we expect: small 6, i f any,

c l o s e enough t o zero t o not be I n disagreement w i t h experiment wi&in i t s e r r o r s . r i z e , t h e Campton s c a t t e r i n g as a function of t above 2 CeV a h w s no s u r p r i s e s o t h e r than what w e can expect from d i f f r a c t i o n from the k n a m t o t a l photon absorption cross s e c t i o n . Below 2 GeV, t h e r e f o r e i n t h e r e s o n m m region, there is no d a t a ,

But i t

should be possible t e make a p r e t t y good theory of angular d i s t r i b u t i o n a d enargy variatlion by considering a s w c e s s l o n of s-chaanel resonances (mny of t h e m t r i x elienrents of which a r e kn5m from t h e study of y p * ap i n the s a m energy region, m h o m ones m y be guessed from t h e quark =del). There i s a l s o a coerputabbe n e u t r a l pion exchange t e r n

The coupling of two photons t o a neutraf pion is

from the no * 2y decay.

A l l these c a l c d a t i o n s can be c h e c k d and controlled bp f i t t i n g the c a l c u l a t e d

imaglinary p a r t of the s c a t t e r i n g t o the n i c e l y m a s u r e d t o t & cross s e c t i o n otot which shows t h e expected resonance b w s i n t h i s region of energy. YN

S c a t t e r i n g of very low Q, v is l i k e t h e s c a t t e r i n g o f radio waves, o r (if q

2

6 Q)

i t depends on the reaction of a p a r t i c l e t o nearly constant e l e c t r i c

and mag;netic f i e l d s ,

This is, of course, gives by two experimental constants,

(obtahed by nreasuremnts i n such f i e l b ) the eharge and the mgnetfc mroent ( r e s t r i c t t o s p b 112 case),

Therefore we expect the Gompton e f f e c t f o r Sow

enou& v, Q to be given e n t i r e l y i n t e r n of these constiurts. should a c t exactly a@ i f they were point p a r t i c l e s .

'.%C?! p ~ r t i ~ l e s

One can cornpute rhc e f f e c t

s e d e l a a ~ s i c a l f yo r fro= the non-relativistic approximation to the S&roedlslger equation with spfn (Pauli equartion) o r again by diagrp a r t i c l e with no i n t e r n a l excited s t a t e s ,

a s s d n g a pure

Sueh a tern is called a Barn t e w .

Thus we write T

(not averaged over spin directions of the proton) a s a sm uv from the Born t e r n and the r e s t from other diagram

B do&natsa a t Tow

The reaaon T

-

V

is t h a t i t has an energy d e n s a n a t o r

due to the i n t e m d i a t a s t a t e A of s i z e (M + v -EA) EA is M

P

(as Q

P

80

if A

%S

0) and we have a leading l / v factor which doesn't appear i n

B the rearalnlng t e r n T

.

& you expect the lestrix e l e m n t of the charge density St i s - the t o t a l charge a d

vation) ; SO aee t h a t

V

Y

is diagonal. thrrs

c xlp (Q) /p>

P a t (7

m

O

2 order Q if x is not

To show Chat the off dlisgonal nratrix e l e m n t s of other

the proton s t a t e . coqonenta of J

a l s o a proton

I

O (charge? ccmserc X / J p> M Y hence i f v , Q go t o zero together we

also go t o zero we look art q

lP>,

x [ ~ / ~Q > .cx~~[o)

3 tnatrix

e l e m n t a go t o zero.

A m r e rjgorow (but har&er t o i n t e r p r e t )

a r w m n t is &even below,

B

We now cowute Ehe lirnit of T a s v ,

9

4

O,

Xf uA is the anonralous

m m n t i n nuclear nametons, the coupling of a photon is yu

For srmll q and tq2

O t h i s l a e r s f i y worked out t o be

F"A ( y u l +5

so

EEecrrontagnetic$4Energy

while the contribution from T

--

fl(ct)

s t a r t s aa uZ nnd thus we have as v

+

O

eL/~

f

v

R

B

To show how

2 2 uA/2H

- e2

o

(47.8)

ssnrall. atore f o r d l y (at leasre f o r the f l tern) note

B

t h a t Iyvt o t a l s. well as T each separarely s a t i s f y the gauge condition qvTBv

-

hence we must have q T

Ci W'

0. NW we can w r i t e T

C

duv

+

.

B

(peQP a+ ( p q l q v b + q

2

it

pM b

kfe cannot solve this by a = +q

--

therefore s

%

peqb, b * o

W

+ pvqy) + W total h e r e must be no poles l i k e l / p * q i n a,b,c (unlike TMv

vhich hae such poles comtog from T

Clearly c

0,

apypv + b(puqv

averaged case a t l e a s t ) a s a power s e r i e s T 2 order qv

-

( f o r the symmetric @pin

-

2

-

1. NW q51 Tliv

+ qv

e

+ order q3

O requtres m

O

/peq because n s 1lp.q t e r n are alLowad.

2 peqa/q with a

$, b .r - o(paq),

c

%

oq2 a r e the only p o s s i b i l i t i e s .

2 a(p*q) and the term s t a r t s out as second d T~~~ can alao be shown t o be of

the same order imd therefore T COmpfon dtscwaion s e c Low, PPlps, Rev,

T~~~ to order 1 and v.

916 1428 (1954)

m$ Gel1

(Par a complete

n, Phya. b v .

96 1433

Forward Compton s c a t t e r i n g from non-relativis b t c Schroedinger equation The equation is (with f i r s t o r b r r e l a t i d e t i c corrections)

3%.

$

h e t d e n t amplitude is

b = Zi, 2

-

tvgi

and

d

is zero, hence the leading term cores from the

contrgbut%ng t o fl,

-+

i-

-

ig x

2.x

gi.

I n the laboratory

- $ + farm and i e -(a 2 /2M)ei*ef

Next we have the terrn a 08 operating i n second order, two

d i a g r m with energy denodnators -v and +v

220

Photon-Hadr~~ lntmctiorrs

-

2

1

+v

r gf) contributing t o f2

2 2

i

F h a l l y t h e l a s t t e m i n (47.9) with rha terln

g.2

x

d

gives

2 which c~labineawith the previous t e r n t o change rhe (1 + U&) to U

therefore have 2

Amp,

-2M &

-

t

ei

*

+

ef

- i2 2

2 - t - b -c v c*(@fx e ) A t

.

Other Two-Current Effects

Other quantities involving T

W

h o t h e r experimental quantity that i n m l v e s our function T

t,lv

4

p/( J ~ J ~ p3 ) ~ /

(not averaged over the spin of the p r s t m , i t involves the a n t i s $8 the byperfine s p l i t t f n g energy i n hydrogen resgon~ziblef o r the 142Q+egacycle line.

I t i s the differeace I n energy i n the gromd

8

s t a t e of a t o d c hydrogen

depending on whether the spZna of the elects011 and proton axe p a r a l l e l o r a a t i parallel,

In non-relativistic approxiwtion

it

depends on the prob&iIity t h a t

2 the electron i s on top of the proton I@(o]f i n the gratrnd etste wave EunctSon. h l a t i v i s r l e a l l y we can write

e , v,

(Rio

= Rydberg,

u , P

~na$neticntawnts of p and

Bohr m p e t a n )

The f a c t o r ( l

+ m / ~ ) - c~o w s

from reduced =ss corrections t o the Schroedinger 2

equation i n getting l+(o) 12;

1 + 3a /2 is a modificaf2on due t o the Dirac

equation,

g, %R..ti" a r e

The other factors

a13 near one and a r e due to higher

order quantum electlcody~&dccorrections.

They have been separated i n to three

factors f o r convenience of discuarp5on. g corns f rorn QED sodif ications of the

221

Pbton-l"iadronInteractions m t i o n o f t h e e l e c t r o n , d i a g r a m l i k e A below.

c o w s From t h e proton r e c o i l d i a g r m of type B i n which the f a c t t h a t the proton i s not a point charge

113

included by using m a s u r e d f o m f a c t o r s . r" c m s

fronr two photon exchmge t e m of t h e f a m C, D.

A l l theere f a c t o r s except y w h i c h depend@on a s y e t m k n m p r o p e r t i e s of t h e p r o t m have been c a l c u l a t e d t o very high accuracy

- f u r t h e r &E is

raetaaured t o a b ~ ~ r d hf iy@ accuracy,

The c o m t m t s l i k e a, 11 e t c , a r e new P knom wels enough t o d e t e d n e r"l t o about four p a r t s per raillion, Theor e t i c a l l y , the deviation due t o AB long a s F r e m i n r ~ = , certain

P

is about t h i s sam order of mgnitude.

we cannot use these accurate m a s u r e m n t s and

c a l c u l a t i o n s t o b p r o v e our knawledge of t h e constants

- or

t o put t h e problem

t h e o t h e r way i f more accurate v a l w s of t b e o t h e r constants becorn a v a i l a b l e that will t e l l

u9e an e l e c t r o m g n e t i c property of

be challenged t o c a f c u l a t e i t , w r i t e f'= f Obvioua3ly t h e proton coupling is t o

+ A,

t h e proton

Let

W

t r ~ acurrents.

- and we would

s e e what is involved, Ve s e p a r a t e t h e caaea

t h a t the photons a r e of low energy mid nromentum fro= t h e cases where they a r e high.

Where they a r e low, binding of t h e e l e c t r o n i n f n i t i a l and inte-diate

s t a t e s e t c . m a t be considered

- b u t here for law v ,

Q, t h e proton a c t s , a s we

have seen, l i k e a. p a r t i c l e of charge and raagnetic mment and thus we c m do t h i s p a r r of the c a f c t l l a t i m .

For defgniteness we do it p u t t i n g i n the exper&-

mental f o m f a c t o r s once f o r each p h o t m and i n t e g r a t e over a l l mraents, c a l l t h i s hl.

Wrlte d

AI f A2.

Now f o r high v i r t u a l moPlentwn e l e c t r o n binding,

even t h e e l e c t r o n =ss can be neglected and we can ionagine the e l e c t r o n and proton t o be f r e e and a t r e s t before and a f t e r t h e s c a t t e r i n g

- thus our two

223

Other Two-Current Efleets c u r r e n t operator

p l J~ )i

a l l t h a t is involived. a r e involved. butions i n AI

T

vv

v

jp>

f o r proton i n m d out of s a m nroanturn ( r e s t ) is

Naturally s t a t e s

X

o t h e r than pure proton Born a t a t e s

Qf course we have already c o m t e d so= high mrnenturn c o n t r i and we s h a l l have t o sllbtraet them, thus we s h a l l have not j u s t

but r a t h e r TPv

e l a s t i c where

-

elastic wv

term alone with form f a c t o r s .

C a l l t h i s TMv'

is c a i c u l a t e d from t h e Born

- TvVe l a s t l c

T vv

Then h2 is

proportional t o d i a g r a m C, D with e l e c t r o n a t r e s t i n i t i a l l y and f i n a l l y .

If Tllvt is

B y

Ear t h e proton

matrix on the proton s p i n i t depends on the s p i n f l i p amplitude

- thus

OR

t h e a n t i s y m t r i e a l p a r t of T

St v

"(proton

2 2

The f a c t o r l / ( q 1 is f o r the two photon propagators.

a t rest),

The f i r s t t r a c e

i s t h e Coonpton s c a t t e r i n g by t h e e l e c t r o n

neglecting the ntanrrctntm p aad mass o f the e l e c t r o n .

The second f a c t o r is the

coupling of ltwo photons t o t h e proton, which we do not know but which we a r e discussing; we need i t s a n e i s y = t r i c

imginary part M

where G

UV

part.

I and C? a r e functions of

be c a l l e d S1 and S2. liv

kre

wrote its

i n the form

q L and v defining t h e imaginary p a r t .

the c o w f e t e s e a t t e r f n g functions of which G

for W

I n Lecture 33

1

and G a r e the itnaginary p a r t 2

That I s , we w r i t e a fona f o r T

except S1, S2 replace G1,

G+and G1,

PV

j u s t 1Pke the above

G2 a r e i m S1,

errbstitution we can express C12 d i r e c t l y i n t e r n s of S1, S2,

Drell and Suflivan, Phys. Rev.

154 1477

Let

Im

S2.

Then by

Qne g e t s (see

61967) aod C ,N. Iddings, Phys

. Rev,

Now we can do m n y things with t h i s t o t r y t o e s t i m t e i t o r c o w a r e i t t o things measurable i n p r i n c i p l e i n ways e n t i r e l y mslogous t o our d i s -

c u s s i a a f e l e e t r o m m e t i c s e l f energy.

For an e x a p l e , we can use Cottingham'a

i d e a of r o t a t i n g t h e Contour on t h e dqo I n t e g r a l from the r e a l t o the i m g i n a r y axes, thus t h e i n t e g r a l s depend only an Sl, S2 i n the negative q2 region. F i n a l l y we can express these coaplete S1,

SZ i n t e r n of G l A 2

( t h e i r iraaginary

p a r t s ) by a dispersion r e l a t i o n , and t h u s express h2 in t e r n s of G1, C$.

Two questions coM up: F i r s t , a r e these wlinom functions, l i k e T 1 0 , ~ ' ) necessary i n the d i s -

1

persion r e l a t i o n o r a r e they a s f o r T2 without constants? q w s t i o n i s almost c e r t a i n l y h o r n , ) s c a l i n g bebavior of Gl,

C2.

(The answer t o t h i s

We can wears because we h a w t h e w p ~ t o t f c

I f t h e r e a r e s u b t r a c t i o n s , t h i s m t h o d is f r u s t r a t e d .

Second, supposing t h e r e a r e no undetterdned functions i n the dispersion r e l a t i o n and b2 can be e n t i r e l y e w r e s s e d i n tern8 of Cl, G2 i n t h e e x p e r l m t a l region; what can we do u n t i l Gl,

G2 is d i r e c t l y measured?

Zt b e c o w s a research

problem t o guess a s completely a s possible t o s e e i n what ranges of =q2, v the &l i2 s ms-ensitive

and use whatever w d e l s o r ideas a r e most r e l i a b l e t h e r e ,

W could t f y t o incorporate a l l that: is Ianm of low energy t h e e r e m and i n t e g r a l s ( l l k e I g2dx

0) t o l i m i t the p o s s i b i l i t i e s ,

At worst, c e r t a i n l i m i t a of un-

c e r t a i n t y can be e s t a b l i s h e d sknce, f o r e x a w l e , 28, which is t h e d i f f e r e n c e of up m d down s p i n partonr h+(x)

- &_(X) (see Lecture 33) cannot exceed the

sum hS(x) l- h_(x) which i s f(x) and i s w a s u r e d , t r a c e of fa T

vv

2 i a e o m way.

W

Generally p o s i t i v i t y of t h e

on any d ; t a p ~ a ls t a t e l i n r f t t h e s i z e of G1,

G2 i n t e r n of W%,

Tfiis problem should bc: pursued

And i f the answer t o t h e f i r s t questlilon i s yes and there a r e d n m

functions brought i n i n t h e fixed q

2

dispersion r e l a t i o n s ?

Then Cottfngbm'a

scheme doee nof work and we s h a l l have t o uge o t h e r mth-ods of a n a l y s i s such a s t h e f i x e d Q dispersion r e l a t i o n s t a abrain expressions on &ich we can apply our p a r t i a l physical understimding of photon badron i n t e r a c t i o n s s c a l i n g , quark nrodel, e r e . ,

- i n conjwnction with a s m n y

stap

- partans,

rules etc.,

we know (as w e l l a s possibly even the n u w x i c a l value of the n-p

that

mss differences)

t o guide us a s much a s possible t o c a l c u l a t e t h i s quantity A2 as accurately a$ possible, a d with an honest e s t i m t e of t h e possible t h e o r e t i c a l l i d t s of uncertainty.

Other Two-Current.Egecrs b c t v r e 49 Other two-current e f f e c t s I should j u s t l i k e t o add a few miscellaneous r e m r k s of s i t u a t i o n s i n

which the double operation of two c u r r e n t s is involved, namely, i n the d i s i n t e g r a t i o n of pseudoscahr =sans.

They a l l present i n t e r e s t i n g q w s t i o n s Of course m t r i x elements

f o r study; t h i s i s Beant raerely a s an introduction.

of one 3 a r e involved LR s i n g l e photon decays l i k e o

-+

1.

ny and we

have already

discussed them. Two 3% a r e obvIousIy involved i n two photon decays Like nQ q0

-+

2y.

-t

2y o r

An honest c a l c u l a t i o n af e i t h e r one of these would be very i n t e r e s t i n g

(one i n which t h e v a l i d i t y of the i l s s w t i o n s is baeked up by considerably more than t h e mere f a s t t h a t the answer agrees with t h i s one experiaent). % a t does Q l (U; + da 2s;) t o g e t SUS say? Use the quark m d e l no 2 (U; da), Q J;Z" Ji;" 1 (ij4 F)/ l L &+ l 2); t h e r a t i o of amplitudes amp nolaw 'Q 15. It say@

-

-

-

- --

-

4-2'

t h a t t h e amplitude f o r n i s high,

8t h a t

-+

2y

is r ( q

-r-

1.0

2y)

+

-

-

48-

f o r q, s o t h e r a r e should be t h r e e tims azi

Experlarentally the widths a r e r(no

width

-

-r-

2y)

+

7.2

1.0 eV, and the; p a r t i a l

The r a t i o is 11140 i n s t e a d of 3:

. 3 KeV.

Qf course, the reason f o r the a b j e c t f a i l u r e i s the very l a r g e -ss difference of r it.

and

FI

2 (mx /m

n

2

m

1/15) and we must be much more c a r e f u l of

This is where SU3 is i n d e f i n i t e

km.

- and no universal way of

&in& t h i s is

F i r s t t h e r e i s phase space: t h e general f o m u l a f o r d i s i n t e g r a t i o n of

an o b j e c t of mss nz a t r e s t i n t o two p a r t i c l e s each of whose mmatura is Q is

where K 1s t h e r e l a t i v i s t i c roatrfx elenromt. space ( i . e . i f we ass-

Xn our carte (2y) Q

m12 s o phase

K i s given by SU3) w r k a agaixlst t h e rt and the d i s -

2 crepancy is another f a c t o r of m /mn worse (i.e. MW /H

rr

'

1/540 i n s t e a d of 3/1).

Plore s e n s i b l y we should w r i t e M in its sienplest: r e l a t i v i s t i c f o m which f o r a pseudoscalar meson d i s i n t e g r a t i n g i n t o 21's of p o l a r i z a t i m el, e 2 , m a n t a kl,k2 is N

a

E

pvap

e

lil e2u

4p *

NW t h e g m s s would be t h a t a l s i d e t e d n e d by SUB. This ~ll?.ans M pea a 3 2 a s Q o r m2 ~ a t e ~ / ~ a -3t e ~, aw2/aq2 = -L %is a t l e a a t m v e s i n IPL f 40

.*.

-

.

the r i g h t d i r e c t i o n , it g h e e

0.4

a:/anZ

i n s t e a d of 3.0

.

This i s not bad b u t (a) what of the r-iaing

discrepancy astd (b) why 2

should a be given by SSU and n o t say a/m o r a/m ?

Evidently we have been dolng

3

too much e m p a r i s o n t o e x p e r i m n t and too l i t t l e thinking. Can we reason o u t f r m o t h e r things we know j u s t bow t h e r r a t e and

rl

r a t e should be compared, o r

bcw e i t h e r might be calculated o r e s t i m t e d absolutely? With regard t o the mchajniaul of t h e decay e v e r s i n c e i t was suggested by Gell-Mnn,

Sharp and Wagner, Phys,

&V.

Jcetters

8 261

(1962) i t is supposed

t o be do&nated by t h e diagrlstro going through an inre-diate

This connects i t t o t h e npy coupling constant, o r through SU

p o r w wson like

3

t o the paeudoscalar

v e c t o r photon coupling constants h general; d e t e d n e d d i r e c t l y f o r e x a w l e by t h e u

-*

ny r a t e .

Again rr

+

nny is i n t e r p r e t e d i n t h e same way a s

The pnn coupUng being k n m one e m campare t h i s with tl t h e r a t i o l"(n

+

nny)/r(n

-*

yy) with good success,

e x p e r i m a t is .l2 ( G a d e g , IPIrys. h v ,

g 501

+

yy

and p r e d i c t

Gel1

(1970)), (For two calculat&lons

corresponding t o different: choices of how t h e coupling constlznts s dewnd on t h e rnasses of the s t a t e s s e e Brown, Muncak and Singler, Phys, Rev, Letters (1369).

22 707

(1968) and Cban, @ l a v e l l %and Torgerson, Fhys. Rev.

The choice made i n the l a t t e r paper f i t s very n i c e , )

185 l754

%%is is i n t e r -

e s t l n g f o r a t f i r s t s i g h t t h e a m r a t o r is order a and deno&aator is order a

2

s o t h e order should be 137 but here m a y n w e r i c a l constants a c c w u b a t e t o o v e ~ e l n rt h i s f a c t o r and taaka the r a t i o n e a r l y 1000 tiws smaller. p i c t u r e t h e r e i s a f a c t o r i n the matrix element f o r where

is t h e i n v a r i a n t mass squared (p+

of the two pions,

+

For t h i s

-

rl + nny l i k e l/(mp2 mn:) 2 of the four-vector slrm p-)

This d i s t o r t s t h e spicsctruar away from t h e simplest Eom toward

Orher Tw-Ctrrrgnr Eflects l a r g e r p r o b a b i l i t i e s f o r l a r g e r rann

2

.

The experiments (Wrarley) a r e s o

accurate t o s e e t h i s e f f e c t , even q u a n t i t a t i v e l y , s o t h e r e can be no do&t of It is l i k e l y t h e r e f o r e t h a t no deep olysteries l i e

t h e mchanisan i n t h i s case,

I n the 2y d i s i n t e g r a t i o n a e i ther. Rowever t h e rt

is

-, of

0 +

the n is

+

+ so

-, charge

0 la a n

t h e d i s i n t e g r a t i o n is n o t allowed, s t r o n g l y ,

W'S

Consider

conjugation (which we think Ss s u r e l y s a t i s f i e d f o r decarys

a t t h i s r a t e ) (Q and no a r e charge conjugation 3.

=quires t h a t t h e a

mis

The G p a r i t y of t h e

3% does present a challenge.

, a- must

+ because

they can go i n t o 2y)

be s m e t r i c , they m w t be i n m T

added t o the X = 1 of t h e t h i r d w e o n y i e l d s only t o t a l L

O o r 2 state. 1, l, 2 , 3 a s

p o s s f b i l i t i e a , t h e r e i a no f = 0, m e r e f o r e the decay cannot occur except by Psospin is, of course, v i o l a t e d try e l e c t r a d ~ a d c s

v i o l a t l o o of i s o t o p i c spin.

s o an eleetrodpna&c v i r t u a l e f f e c t (order a i n rnatrix e l e w n t ) i s involved and the? r a t e is a f order cr

2

510

rl

swaqed by what S t would be i f

2y can compete with i t r a t h e r than being c o q l e t s l y Q -*

371 were a s t r o n g i n t e r a c t i o n ,

(The d a t a on

rr decays is Branching r a t i o X

* 1.1

23.1

'&e change i n X s p i n can be bX G parity

.

. ;

Hence the f i n a l s t a t e of t h r e e pions must have

can get ao estimate af t h e r a t i o

I part violates

O, 1, 2 but only t h e BP,

r(n

3n0)/r(n

+

a+71-n').

P,

= 1. From t h i s we

Conrbining t h r e e

s t a t e a of i s o s p i n ,none we can get f o r t h e t o t a l I spin:

tric

2,2,1,1

Pf we suppose t h e 3% a r e i n t h e i r l w e s t space wave, since they a r e Base we w i l l have f spln 3 o r 1;

by proper use of space s t a t e s t h e skew-S

S

wave m d sy-tric,

EM p e d t e only t h e l (but

t r i c X = l s t a t e s could c o w i n ) ,

Foa t h i s s t a t e

r(n

; .

3no)lr(n

-i

n'n-no)

= 3/2.

Sf

t h e r e is s m e skew s y a m t r i c

( m m e n t m dependent) space s t a t e the o t h e r Z = 1 can come i n reducing t h e 312; s i n c e i t corns i n with h i g h e r angular wmtentm i t i s probably s a t a l l e r s o t h e r a t i o m y be f a i r l y c l o s e t o 312. Bow do we c a l c u l a t e t h e r a t e

f t is 1.3 e x p e r i m n t a l l y , i~ -s

3n?

%ere does t h e i n t e r m e d i a t e photon

a c t ; can w e guess whl& i n t e r m d i a l r e s t a t e s a r e mast likely t o be i q o r t a n t ? A quilntitave e s t i m t e f o r t h i s r a t e is a problem t h a t no d i s c u s s i o n o f t h e

e f f e c t s of v i r t u a l s e l f e n e r e photon a c t i o n can f a i l t o arention.

Hypotheses in the Parton Mode

Lecture 50

We should now l i k e t o discuss what we can say about what t h e products

R would look l i k e i n deep i a c l a s t i c e e a t t e r f n g e

+p

+

e C X,

There a r e s o w

wasurerolcsnta f o r c e r t a b dcSef?lnite fSnal s t a t e s X f o r satall energies and low 2

q

of t h e v i r t w l photon (see Berkelmn, 1371 CorneZi conference),

Most

of these can be w d e r f ~ t o o dEron d t r e c t e r t m s i o n s o f our theorfefir f o r photon (qZ

0 ) reactions.

We have already d i s e a s e d pion production from v i r t u a l

pion exchrange a s y i e l d i n g info=tfon

on t h e pian f o m f a c t o r .

Xn addieion

c praductian has been s t u d i e d from v i r t u a l photons, wPth no surprises gives a f a i t account

- VBM

- s e e our theory d%scweed i n Lectures 16 t o 21 where we

2 2 = r e l y have t o replace kout by q i n the equations t h e r e yieldling a f a c t o r

m 2 l (mp'-q2) P

2 (q i s negative) r e l a t i v e t o t h e

-

O ease.

Zt i s necessary a l s o ts m k e an a s s m p t l o n of how the l o n g i t u d i n a l l y

polarized photon (possible when q

2

f 0) couples i n r e l a t i o n t o the l o a g t t u d i n a l l y

s a s s m d t h a t these a q l i t u d e s a r e r e l a t e d by a f a c t o r 2 t h e e x t r a f a c t o r q /m

P

is anansat z made because gauge

invariaace require8 chat J longfrudinal vandahea aa qL

-t

0, Thie m y be v a l i d

f o r n o t too l a r g e qZ, b u t of course i f q2 becomes r e a l l y vary l a r g e the

229

230

*

assumptions about small p + p production may begin t o f a i l . What happens a t l a r g e q2, and how i t ties on t o small q2, i n every case

€or very

v w i l l be our present concern.

W e s h a l l have t o be guided by

theory and I w i l l take t h i s opportunity t o review t h e parton model and some assumptions t h a t can be made about i t .

W e s h a l l list assumptions t h a t we can

- without today being s u r e of which a r e r i g h t and whicb wrong - j u s t t o

make

see what t h e i r consequences a r e with t h e hope t h a t experiment may later make the s e l e c t i o n (e.g.

are charged partons quarks?),

Therefore i n our list

some assumptions w i l l (perhaps) be i n c o n s i s t e n t with others.

The assumptions

w i l l come from a mixture of t h e o r e t i c a l guesses and known experimental

facts so-

- so

one might be warned t h a t i f a p a r t i c u l a r asswnption n e a t l y explains

experimental f a c t i t may not r e a l l y be a s i g n i f i c a n t confirmat

assumption might have been made with t h a t f a c t i n mind. e f f o r t w i l l be made t o derive one assumption from another n o t independent.

Finally, l i t t l e

- they are c e r t a i n l y

T h i s w i l l t h e r e f o r e unfortunately not be a m a t h e m t i c a l and

sound system, but r a t h e r a lengthy " i n t u i t i v e " o r physical discussion, For a good discussion along the same l i n e s see J . Bjorken's paper i n the 1971 Cornell Conference. General Framework W e suppose a s i n f i e l d theory, a wave function f o r a s t a t e can be given

by giving t h e amplitude f o r finding various n m b e r s of f i e l d quanta, o r partons of various momenta.

I n p a r t i c u l a r we discuss the wave functions of s i n g l e

p a r t i c f e s (eiomrttimes two p a r t i c l e s i n c o l l i s i o n ) with extremely l a r g e momentum P i n the z d i r e c t i o n (P

-c

The wave function i s being described i n Fock

QD).

space giving f o r a s t a t e the amplitude

etc.

where JIo is t h e amplitude t o f i n d no partons (usually zero); ql(pl)

is t h e

amplitude t h e r e is one parton (of such and such a type, an Lndex we are omitting)

231

Hypothese$in the Parton Model A i c h has mnrentm pl; J"Z i s t h e a w l l t u d e t h e r e a r e plS p2 e t c .

represent the vaeum s t a t e and a

p.

P

partons of Qtolltenta

For example l e t /VAC>

This can be w r i t t e n i n o t h e r ways. rL

WO

t h e operator c r e a t i n g a parton of m m n t u n

= P* /VAC> where

Then we can w r i t e the wave function e t a t e

is s o m function o f the c r e a t i o n operators,

Then we make the following assumptions : A&, -

The amplitucte t o f i n d a l a r g e

)&I on any parton f a l l s rapidly with

such Char e f f e c t i v e l y we c m i n f i r s t approximtion consider a l l Pi a r e f i n i t e (as P

+ W),

AZ. Tlne '*wave function" f o r Longitudinal m m n t m of order P, i . e . g t t = s

This r e q u i r e s some conrplicating r e m r k s t o m k e its d e f i n i t i o n c l e a r f o r t h e r e is, a s P

+

-, an

ever increasing c s n t r l b u t i o n t o $I for small

More p r e c i s e l y consider t h e density ntatrix. t o f i n d n partons of m e n t a p1 t o pn,

Let *m(plpZ...pn)

X.

be t h e m p l i t u d e

Then f o r exaraple t h e density f o r one a t

k is

That t h i s cfepends only on kL and x = kL/p when a s s w t l o n we want t o make

- with

p a r t i c l e density matrix [ l i k e dpl,,.dp,

1

X

is f i n i t e as P

a l l its generalizations.

/$i(P1pP2s * *kg .P,)

p(k,kl) depends only on hl,k;,kL/P;ki/P

wave f w c t i o n i n p

It is a l w e t but not q u i t e X%

-

is t h e

E.g. t h e one

(plp2. * k t .*P,)

e t c . , when

1

X

is f i n i t e .

6 (pi-kl) 6 fpJ-k2) )

Again t h e two p a r t i c l e density (the expectation of behaves likewise e t c .

$ ,

+

t'ki same a s saying

PLI/P, has a d e f i n i t e l i m i t .

the

The wave function

is a function of a l l of t h e mracmta including those of f i n i t e mmntuan The s c s l i n g d o e s n t t work f o r those

(which we c a l l weea).

- in

m a n a w e r of p a r t i c l e s r i a e s wlth P s o the wave f u r c t i o n JI

f a e t the f o r any fixed

n f a l l s with P ( l i k e a power of P) but t h e ' k e l a t i v e wave function" e.g., r a t i o i n which only one p a r t i c l e f o r a f i n i t e n a b e r ) is mved depends f o r finire

X,

A3. -

p mly

on

X,

pL.

I n the wave function the a q l i t u d e t o f i n d

a

232

Photon-Hadron Interations That is t o say again t h a t t h e

density matrix, e.g.

the density Ear finding p a r t i c l e s with f i n i t e (e.g.

behavior as P values of pL have a d e f i n i t e l i ~ t i n g nmbex o f such wee

-3.

m,

4-4 GeV)

and the expected

partons is f i n i t e e

& 4 .- To have continuity between A3 and A2 the mm n a e r of partons of

a given type

z&

The n d e r Ear pZ negative f a l l s o f f rapidly 80 t h a t f o r p, =

as B finite x

X$?,

negative t h e r e a r e no partane; although f o r pZ negatlve and f i n i t e t h e r e a r e some (a fixed =aunt

f a l l i n g rapidly with negative p Z ) .

(As a t r i v i a l e x a q l e of the kind of bhthavior envisaged i n the wee region where a* c r e a t e s s p a r t i c l e of consider a wave function l i k e exp (I: C ~ < ) [ O > Er l o n g i t u d i n a l mooentum h and Ck v a r i e s a s $ . / ( w - ~ ) Y ~ / ~ with W say,

-

- \fi

and a = constant.)

A%, The bE?hwior of the wees is nearly (as P of t h e d i s t r i b u t i m of t h e f a s t ( f i n i t e

X)

-3.

partoas.

a

completsly) i a d e p e n d ~ t

This again is connplicated,

I f we s t r e t c h o u t t h e v a r i a b l e pz by, f a r example, defining y

+ pz)

so f o r f i n i t e pz, y i s f i n i t e ; f o r f i n i t e n,y is &nZF

We have p a r t i c l e s a t every y fro= f l n i t e

0

Ln (

; . 03

f i n i t e RnZP-y ( f i n i t e

X)

+ 1 GeY2

+ enx.

t o Rn2P.

Rn2P

We have d r a m a graph oE t h e mean rider of p a r t i c l e s i n dye look near f i n i t e y, P

\/p:

I f we

we s e e the behavior of t h e wees; i f we look a t

we s e e t h e behavior of t h e s c a l i n g f a s t ones;

in

between i s a plateau with a E i n i t e density o f partons so the mean a m b e r of partons r i s e s a s EnP, It i s easy t o m d e r s t a n d the density but how do we mderstand the wave function?

"Phis gives t h e amplitude far every configuration, whi& &B a s e t

Hyporhesgs in rhe firton Modd of values of y f o r partons present

Ln2P

0

Y

It behaves

Bow does t h i s w l i t u d e vary with t h e p o s i t i o n of the data?

l i k e a wave funetian f a r a f i n i t e one d%mnsional drop of llqrtfd of thickness Ln2P. The a a p l i t u d e is l a r g e only i f p a r t i c l e s a r e rtlazrs o r l e a s e v e v h e n ? , with a uniform density except near t h e surfaces a t &n2P and 0 .

The behavior

a t one s u r f a c e is not s t r a n g l y influenced by what the configuration is a t the other: s u r f a c e

- they a r e i n s u l a t e d

from each o t h e r by the long

c?,

Rn2P is

never r e a l l y very l a r g e ) i n t e r w d f a r y plateau. Mathematically we a r e discuhising the prolution of W+ = Eiy f o r a rstate of

If

fined Pz (but we do n o t knnr H, of course). (e.g. l i k e

E pz a;ak

e t c . ) we want?

eigeovector of the operator W

iy

m

- - z,

F*.

qzis t h e momentum operator Cansider then t h a t

f o r a s t a t e of masn H.

H

@

is an

Now as P

+

m

2

P+% IZP) we eaa consider t h e lirnlt

(E

-

(ZPW)$

nZiy

(li1.1)

so we a r e Looking a t eigeavectors with fixed efgenvafues o f t h e operator PW a s P

+

=.

Wi,

axe a s s u d n g t h a t i t loolrcs a s i f , 8s P

d i s t i n c t l i n t i t , e x p r e s s i b l e i n x = pC/P and

is t r o d l e on t h e s m s l l mnrentm end,

p,.

This would be n i c e , but t h e r e

The equation is l i k e a cadieade, l a r g e x

genkrates staaller x through i n t e r a c t i o n t e m (I%& c o s e e ray showers).

t h e o p e r a t a r PM has s

+

turbulence equation8 , o r

Smiler matenta p i l e up m t L f new ptzenomna s e c s i n t o

change the equations (lLke viscoajtty i n turbulence, o r i o n i z a t i o n l o s s i n cos&e rays) t o f i n a l l y h t e d n e t h e wee

X

(EInlte pZZ behavior,

(The approxi-

aaatim I n showing t h a t PW depends only on x is wrong, f o r e x m p l e we c m no

longer: w r i t e

pZ,)

But by t h a t time the "'shower fkr f u l l y deveiopedf'

and t h e behavior of t h e weaa (except f o r n o m l i z a t i o n wee#) is indepenrient of t h e way i t e t a r t e d a t f i n i t e

- t o t a l strmgck

X,

of

(In the wee region

i n t e r a c t i o n e n e r g i e s a r e comparable t o k i n e t i c energies.) l'be behavior a t t h i s md Is a e o l u t i o n of t h e equation

W=O (Note the e ~ a s i a nof P , )

Tn general the operator W dioea n a t have a zero

Photon-MadronInteractions eigenvalue i f a l l boundary conditions a r e used condStion of f i n i t e n e s s a s pz equation H$

m

E@ f o r E

say the condition a t r =rely

J.

+ m. I l t

- b u t here we r e l a x the boundary

is l i k e s o l v i n g t h e Schroedinger

O when i t does not have t h i s eigenvalue by relaxing + m

and thus studying open s c a t t e r i n g s t a t e s spproxi-

t o which t h e r e a l l a r g e r behavior will. have t o be attached

- here we

mmt ultilnatelg r e a l l y a t t a c k t h e f i n i t e x s o l u t i o n of ZPW* = mL$,) It can b e s h o w t h a t s i n c e equation (51.2) is i n v a r i a n t un&r a boost

Lorenta t r a n s f o m a t i o n i n the z d i r e c t i o n by v e l o c i t y v, c a l l E i n which a l l l a r g e pZ a r e multiplied by f , t h a t JI c m be of t h e f o (so boosting does n o t change 9 , only t h e n o m l i z a t i o n ) .

This m a n s t h a t f o r

f i n i t e but smsll x the p m b a b i l i t y of one parton a t n v a r i e s a s nZ8dx/x. i n A4 corresponds t o B

lowest B s o l u t i o n we asstheory).

The

O ( f r o a e x p e r i m n t , not

Other s o l u t i a w e x i s t f o r higher B and the general s o l u t i o n i s a

l i n e a r conibinatlon of these d o s e c o e f f i c i e n t s a r e d e t e d n e d by how they f i t on t o (51.1).

S;

had hoped t o g e t a f i e l d theorqr i n t e q r e t a t i o n of b g g e

theory t h i s way but I have n o t c m p l e t e d t h e analyeis. A6. -

The d i s t r i b u t i o n of the wees is the s a m f o r a l l hadrons; Wng

a bold assumption p a r t l y guided by e x p e r i w n t ,

This is

o t h e r things i t m a n s

t h e weee a r e neutral. t o i s o t o p i c s p i n ; the wee8 f o r t h e proton look j u s t a s they do f o r the neutron.

f i e d i f f e r a c e c m only corn fro= a higher & and

hence f a l l s i n amplitude r e l a t i v e l y as P-@ (B t r i c w i ll with experiment

f;n

7

0).

I h e assumption t h a t the

Be made, (P think i t leads t o d i s a g r e e m n t

the expected r a t i o s o f n's t o K ' s f n c e r t a i n experilacmts)

f o r we imagine i n t e r a c t i o n forces a r e e f f e c t i v e i n d e t e r d n f n g t h e wee diat r i b u t i o n and such forces a r e not $Ug i n v a r i a n t .

(It s t r i k e s W*

nte a t t h i s mannear, t h a t s i n c e t h e wees a r e d e t e w n e d by

0, i , e , , a s t a t e a f zero =ss squared, and t h a t s i n c e pions have a small

nraea, t h e s t a t e of the wees m y be approximtely only pions (with kaone, of l a r g e r mass lauch reduced, hence l a r g e SU3 breaking,)

Knam pion i n t e r a c t i o n s

(perhaps described by intremediary p nzesans) might p e d t a s o l u t i o n of W$

I n t e r m a p p m p r i a t e pion base s t a t e s .

If you do t h i s 1 would suggest i t

mi@t be e a s i e r i f you work a t f i r e t with t h e

S

t r i c (inrpz) wee d i a t r i -

bution corresponding t o two f a s t h s d r m s c o l f i d i n g , r a t h e r than t h i s onesided, one-particle d i e t r i b u t i o n , )

0

23.5

Hypotheses in the Parton Mode1 A6 is a c t a w l e t e l g obvious from f i e l d theory

The a r p s u q t i m

- far

there might be so=

long range d i r e c t e f f e c t of the f a s t partons on the slow

ones i n principle.

The choice i s guided by expctrimnt (which shows t h a t the

right-mving products of hadronic co3LlisPons depend only on the right-moving i n i t i a l collidkng p a r t i c l e and not on what i t c o l l i d e d with. a s a m p tfan is wed s e e B1, a i d J, k n e c k e e t a l , Fhys, Rev. The physlcal a s s u w t i o n B 1 t h a t we

m&@

To aee how t h i s

,188 2159 (1969).

l a t e r says specifically t h a t there is

no such long range e f f e c t .

G. Continuity

demands, s i n c e the wees a r e adjacent t o the s e a (plateau

region) t h a t we a l e o have t h a t the s e a (e.g. =an n m e r s and corselationa of

rnI?kY2 where y

depends on the q u a n t a numbers

(angular mmntlmt, isospin, strangeness) c a r r i e d by the gap.

E,g, suppoee we

have a proton e t a t e and we ask f o r c e r t a i n partons s , b ,c, f o r y

y

1 and others

below y2

?J2

Perhaps t h e i r strangeness is contributim S

yl

+ 1, t h e

- 1 across t h e gap.

O s o there is a

e n t i r e proton h a s S

It 2s evident t h a t t h i s "quantura a d e r 8

carried across the gag" defined a s the qumttas a d e r s of the s t a t e d n u s Phase t o the r i g h t o f t h e gap (a,b ,c) is j u s t the sm of those t o t h e l e f t (S, t l

.

This more c q l i c a t a t d aethod of exprwslon ts zm a n t i c i p a t f o n of the

s a m i d e a f o r distr&b,butioaswhen two hadrons a r e c o l l i d i n g ,

Then i t is the

q u a n t m riders of the h a d r m taovgng t o the ri&t ntlnw the t o t a l qwntran a m b e r s of the partons t o t h e rilJht of t h e gap, A s s q t i o n A8 i s not stated, c l e a r l y , chmlgc?d.

f t is used i n two cases:

l)

We have t o say haw the gap by is

&@ion a,b,c and s,t a r e both s t a t e d

completely ancl t h e gap vldents s i ~ l became y P Increases; hence dy = Rn2P mn td the m l t u d e Earls a s . '-P

This was used i n ernslyzing the proton form f a c t o r

i n l e c t u r e 29, f a r e x a m l e .

2 ) The gap 1s i n a plateau.

there i s a l a r g e a t r e t c h of plateau,

On one s i d e or both

Here a,b,c i s flxed and s , t e t c . , is

m y t h i n g a t a l l over a rJ-ide r a g e of y up t o the o t h e r boundary (order Rn2P

236

Photon-Hdron Interactions

away) by i e k e p t fixed as %n2P rises. x,l

X

2

a r e the

X

ProbabilLty goes as (x2/i17

where

v a l u e s a t each end af the gap (used for aeyntptotle b e h w i a r

of deep l n e l a s r l c scattering n e a r

X

=

51,

Hadron-Hadron Co isions at Extreme Energies

Altlrou&tt our subject is photon-hadm i n t e r a c t i o n s we s h a l l review the a s s w t i o n s made in describdng hadron-hadron c o l l i s i o n s A energies,

+B

a t extreme

kle f i r s t leave out e l a a t i c s c a t t e r f a g and d i f f r a c t i o n dissociation

and aim t w a r d t h e l a r g e p a r t af t h e cross s e e t i o n where s e m r a f p a r t i c l e s a r e exaltted A

+

8

-+

C

+D +E +

---.

Far a hard c o l k i s i a n suppose the matenta of A, B a r e PA, PB reepectfvely

in the z d i r e c t i o n

- for e x a q l e

take eenter of mss P

f i n i t e z-vefociry v t r a s t s f o m t l o n from t h i s PA We only work with FI\,FP o r P

i .

fP,

=,

The a s p p t o t i c a l l y inco&ng wave f w c t i o n w i l l be, of course, (i.e.

before "interactdon'"

),

so= kind o f product wave functicn of A of mmncunr PI

t o r i g h t a d B of =matrun PB t o l e f t ,

Technical problem a r i s e here.

Ia f i e l d

theory t h i s e m n o t siarply be a product of the wave Emctions of each, p a r t i c l e t h a t we have been describing f o r t h a t Ss not m i q u e (for exmple suppose A contains a Fernion parton a t =maturn p, and B arcto contains one of t b s s a m

kind a t t h e s a m momatu~a,but there cannot be two In the f i e l d a s they a r e Femions).

Thw %f A is represented aa a creation operator :P

an t h e vacuum,

+A

which c r e a t e s a l l our partons, and B by :F we can define the

F:IVAC>

i n c o d n g asymptotic wave function a s

**

m e r e i s so=

trouble i n the wee regfoa where PBFa do not c o m u t e (note

*

t h a t no c r e a t i o n operators f o r f a s t partons p ^ xPA appear both i n FA and F; because A and B a r e m v i n g i n opposite d i r e c t i o n s ) ,

Actually t h i s is only

technical b e c s w e we only want t h e s t a t e a f t e r the i n t e r a c t i o n .

The problem

would a r i s e only i f we were q w n t i t a t i v e l y c a l c u l a t i n g the i n t e r a c t i o n s ; b u t now

WC

wish t o t a l k about how the wave function looks a f t e x i n t e r a c t i o n hence

say a f t e r "2nteractfon plus correction f o r overlap i n defining the i n i t i a l matate".

%C o w r l a p a f f e c t s only t h e wees, but we s h a l l as@=

the i n t e r a c t i o n

a f f e c t s only the wees a l s o .

(Ely i n t e r a c t i o n we

t h a t although P ;

/VAC> a r e both eigenfunctions of H $> =

/VAC> and :F

* * / W> is n o t .l

t h e rzffecte o f the f a c t

/

F o l l w i n g we s t a t e t h e a s s q t i o n s we s h a l l d

E /$P,

e regarding t h e i n t e r a c t i n g

wave f unc t ion. 81, -

Partons i n t e r a c t anly i f t h e i r r e l a t i v e four-mntm

-

w s m i n g they have s o w f i n i t e m s s of order L CeV.

is f i n i t e ,

This is equitrelent t o

-y f o r p o ~ i r i v ep=) t h a t partons 1, 2 i n t e r a c t only i f t h e i r r e l a t i v e y value , y l ~ Z is of order one o r smiler.

(I use L GeV off e t c .

f o r t h e general energy values a t which i n t e r a c t i o n s f a l l

Z suapect i n aeveral a p p l i c a t i o n s even a s m l i l e r value (e.g. pi,

average) m y be c o r r e c t , althouhJh possibly l a r g e r i n so=@ c i r c m t a a c e s

-

Z t of course cannot be defined p r e c i s e l y wethaut a q u a n t i t a t i v e theory.)

We umae t h i s a s s m p t i o n t o gee a t the wave function ( i n t e r n of parcon distributionsl) f o r the outgoing f i n a l s t a t e a f t e r i n t e r a c t i o n ,

The d i s t r i -

bution i n y of partons f o r the i n s t a t e s A and B have ranges of y snrall t o Ln2PA m d

- Ln2PB t o small respectively.

( e f f e c t of i n t e r a c t i o n ) over a range by

We put them together smearing things .r

l, %is srnetlring near y = O j o i n s

the p o s l t i v e and negative y regions (fro= G and B respectively)

.

Since these

regions were the s a m far both A, E (see A7) t h i s can be done most mairnply by

Wradron-HadrunCollisions st Extreme Energies j u s t extending the c o m a p l a t e a u region f r o a m e t o the other. p o s i t i o n of t h e CM Leaves no t r a c e then,

The exact

as a general p r i n c i p l e .

We a s s m e t h i s

Longitudinal t r a n s f o m t i a n s with a velocity v not too eLose t o c feave t h e (Assumption due t o

distributions of such p a r t i c l e s e s s e n t i a l l y unchanged.

C.N. Vang.) t n our a p p l i e a t i a n t h e p a r t i c l e s a r e partons, t h e t r a n s f o m t i o n a l t e r s t h e p o s i t i o n of the o r i g i n of y by &nf

1 Rn

1.t.v F

[a f i n i t e m u n t ) and tbe

assumption says t h e distribution should look t h e s a m .

Hence the s ~ a r l n g

j u s t has the e f f e c t o f extending the p l a t e a u region of A ermothly back i n t o B.

Me tni&t ask kif the srnearing of hx = X changes t h e dilstribution of partantl f o r y near Rn2P where they were determined a n t i r e l y by A .

(This region Is

c a l l e d t h e A fragmentation region, near -Rn2PB t h e B f r a w n t a t i o n region.) But near A the d i s t r i b u t i o n already s a t i s f i e s (in t h e sense t h a t a l i q u i d surface 1s nearly independent of what goes on deep below i f forces extend over f i n i t e d i s t m c e s ) the wave e q w t i o n , s o we as@-

It

it3

not chrmged,

Therefare we irnagincz t h a t the f i n a l r e a l hadron p a r t i c l e e c o w from the d i s i n t e g r a t i o n of an ""original" parron s t a t e which has the following p r o p e r t i e s .

fragmentation region (y near En2P

, i,e,

for

X

< 0 ) and l i k e

Thus t h i s is campletely described i n t e r n of t h e wave function8 f o r

s i n g l e p a r t i c l e s described i n a s s q t i o n s A 1 t o A8.

We emphasize again t h a t

our a s s m p t i o n s a r e not independent,for e x q l e , f o r B 3 t o work the p l a t e a m from each p a r t i c l e must be t h e @ a m a s A7 stays,

@e a r e n o t t r y i n g t o develop a

l o g i c a l system of asrsusptioas, but J w t s t a t e a n m b e r of mutually conraistent (or possibly i n c o n s i s t e n t

- s e e quark

assumptions l a t e r on) i d e a s ,

The p i c t u r e we a r e developing i n B3 is a wave fanetion l i k e a l i q u i d i n t h e varPabSe p wSth s u r f a c e s A, B a t which the d i s t r i b u t i o n s a r e unique but an f n t e r i o r plateau o r s e a region which s e p a r a t e s them,

%ey can be well

separated by taking P l a r g e enough, f o r they a r e separated by Rn2P is u n i v e r s a l ,

Further the r e l a t i o n a o r c o r r e l a t i o n s from one place i n y

t o another have a f i n i t e y r m g e of order one t h i s s e a i s l i k e a Mtlrkov chain weorrelated.

- which

- s o t h e general behavior

- with enougb separation i n p

in

thSngs beconre

Many obvious p r o p e r t i e s expected f o r such a &aSn can be

expected here but they w112 n o t be a l l e x p l i c i t l y s t a t e d ( f o r e x a q l e , t h e p r o b a b i l i t y t h a t t h e r e w i l l be no partan a t a l l of a given type i n a rsrnge by gwa a s e x g ( 4 h y ) f o r stollte C f o r l a r g e enough Ay, e t c . )

The ward "appropriate" 5n defining the wave fuslction i s purposely vague

f o r I a s n o t s u r e h e t h e r I am describing the f i n a l outgoing wave function a f t e r i n t e r a c t t a n when a l l the p a r t i c l e s a r e eeparating o r one i n btlrween t h e i n i r j l a l and f i n a l ,

I have not c l e a r l y reeolved aty c o n f w i o n on thLs toattar

-

but as Z oalyuste the function q u a l i t a t i v e l y i n a m n n e r described i n t h e next l e c t u r e Z have not had t o c l a r i f y i t .

On drawing the p plotcl t o deracrlLbe the partoa wave f m c t i m of asrswptions

B2 and B3 we a s a m d the wee region near y

1

O where A, B tnteract,as being

completely healed over and j u s t a srnooch continuation of a pfrlteau through y

O.

TI.li8 is physically what 1 want and lea& t o 62 (next l e c t u r e ) with

Hadron-Hadron Cotlisions af Extreme E~ergr'es u n i f o d t y i n t h e y defined there f o r physical p a r t i c l e s rsnd i n a c c o r h c e with invariance under f i n i t e v e l o c i t y t r a n s f o m t l a n s which applies t o p a r t i c l e s , But t e c h n i c a l l y t h e cume in the y space y = %n d g h t s h w a b m p near y = 0, a b w p which m v e s when we m k e a f i n i t e Lorenrz transfo-tion,

which is f%ne s i n c e wave f m e r i o n s need n o t be r e l a t i v i s t i c But i t a w e be a b a p

i n v a r i a n t s (X thank F. M e r r i t t f o r p o i n t i a g t h i s o u t ) . (90 c ~ n s t r u c t e dt o

have no p h y s i c d e f f e c t a s a b m p i n the f i n a l r e a l hadron

d i s t r i b u t i o n C2.

It is a "theoretical a r t i f a c t " due t o earelesaness i n finding

t h e ri&t no-lization

and d e f a i t i o n of v a r i a b l e s f o r the wave functian.

Rn

r e s u l t cannot r e a l l y be s m o t h i n t h e s c a l e of y would n o t be s m o t h (near y

0) i f Z had a r b i t r a r i l y chosen t o w e

f a r dyldy ' i s not c o n s t m t

y ' = Ogn

The

.

Latcture 53

We naw go on t o describe what t h e products &&t

look l i k e i n hadronic

m l l i s i o a s (astill leaving o u t d i f f r a c t i o n diseociation).

We have, of course,

no q?lantitative way t o g e t from the wave function described i n p a r t m e t o t h e wave f m c t i o n described i a outgoing r e a l hadrons, t h a t i n y space the r e l a e i o n of: p a r t m hadrone

+

partons described i n A .

-+

But we s h a l l s i ~ l p l ya s s m e

hadrons is muck l i k e t h e r e l a t i o n

Later on we s h a l l have t o describe t h e

produces expected from s t a t e s wfifch m l i k e that: i n B3, have gaps i n them, f o r e x q l e , a s t a t e with J u s t two parteas a t opposite ends of the y w a l e , separated by 2P, W assume ( t h e "conrplement" of B3):

partons

It h a s been ptoposed t h a t t h i s "sea correspondmg t o a parton gap" be

i t s e l f a gap.

This w i l l not be i n c g n s i s t e n t with what we w i l l say next (C2),

and which we w e i n hadron c o l l i s i m s ,

But i t does n o t seem reasonable by

i t s e l f t o ale i f liadron~m k e a un;ivereaI s e a a s 82 supposea

- f o r Z think

that

m a n s t h a t i f there were any d f s t r i b u t i o n a f two lmpbs of hadrons with a gap between, they would w k e a s e a of partons, s o i f there i s no s e a o f partons t h e r e is no separated l-e any r a t e i t i s

n?y

of hadrons, but there must be a hadron s e a ,

At

s t r o n g b e l i e f t h a t there is i n f a c t such a s e a Sn t h i s case

m d not a gap i n hadron m m n t a corresponding t o t h e gap i n parton w m n t a , A63.

A. CSsneros points o u t t h e two outgoing I m p s c a r q l n g oppoeite

+

hadronic quantusn n d e r s (in the case e e-

-t

hadrons) would generate a dipole

s t r o n g current ulhich would tend t o r a d i a t e s o f t e r hadtrons i n t o Low x,

To pre-

vent t h i s r a d i a t i o n becorns fnereasing1y m r e d i f f i c u l t a s t h e energy increases, (as f o r m - b o d y exchange r e a c t i o n s ) .

Gny exclusive m - l u q p r o b a b i l i t y w i l l

f a l l a s a p w e r of energy r e l a t i v e t o tha t o t a l inclusive reaction i n which the r a d i a t i o n , generating an i n t e r n a d l a t e plateau, is penair fed, Now f o r a wave function a s i n B3 we can iraagine the various partons d i s i n t e g r a t i n g more o r Less l i k e i n Cl but n o t r e a l l y independently, tholse a t the ends of the y raage deternine the. hadrons t h e r e , and those i n t h e c e n t e r a f f e c t i n g the hatirons i n the center, but i n a universal manner independent of y i n t h i s region,

Thus

W@

again g e t a hadron d i s t r i b u t i o n l i k e t h a t i n C 1

w i t h t h r e e regions, but the plateau may be a new and d i f f e r e n t d i s t r i b u t i o n

but a s e a nevertheless,

( m e t h e r the two plateau regions, t h e one i n C1

corresponding t o an i n i t i a l parton gap, and t h i s one f o r wave function B3, a r e t h e s a m o r n o t is a d i f f i c u l t problem I have not y e t been able t o decide, We s h a l l

call t h e assamptian t h a t they a r e equal C6, s e e l e c t u r e $5.)

This

a e s m p t i o n can be m d e by J u s t repeating t h e wording of Cl j u s t changing t h e naolcs of t h e s e a , o r i t can be put i n m o t h e r t o t a l l y equivalent way.

H~dron-HIrdronCollisions at Extreme Energies There i s a possible confusion here bemeen t h e "'initiaLWwave

Remrk;

function of C1 and the "appropriate" wave function of 83.

C l i n ece-

c o l l i s i o n s is j u s t a f t e r t h e i n t e r a c t i o n with the photons

- t h e parron

p a i r is j u s t t r e a t e d , like a

** ac a

There s t i l l remains tlrne f o r i n t e r a c t i o n s (via t e r w

i n t h e Wm%Itonim) t o a c t before we reach t h e partan representation

of t h e "final" h i t i a l s t a t e fumction i n t h e sense of B3,

- i.e.

before we reach the '"appropriate"

wave

%%is i n t e r a c t i o n converts t h e i n i t i a l f a s t

parton i n t o two o r Illore, and these aggan a r e broken up e t c . , i n a c s c a d e fashion naitkfng profound changes, f o r e x m p l e by f i l l i n g i n the gap i n tfie low pZ region md c r e a t i n g so= s o r t of parton plateau i n t h e f i n a l outgoing s t a t e ""appropriate" t o t h e i n i t i a l s t a t e CL, The reason no such extensive m d i f i c a t f o n is raade i n going front the i n i t i a l s t a t e i n a h a d r m c o l l i s i a n t o t h e f i n a l appropriate s t a t e B3 i s t h i s , "Prhe d i s t r i b u t i o n of the f a s t s a t i s f i e s W*

( n o n e e e ) partons i n the i n i t i a l s t a t e already

EJI s o l i t t l e disturbance is worked t h e r e by t h e m i l t a n i a n ,

Only i n t h e w t u a l l y overlappfng wee cegicras does the f u r t h e r a c t i o n of M linodify t h i n g s ( t o slnooth o u t t h e plateau).

This f a t r u e n o t became d i s t a n t partons have no e f f e c t , b u t r a t h e r b e c a w e

they have a rmiversal a f f e c t ,

Eere (and i n Cl) y c m be more p r e c i s e l y defined

f o r t h e hadrons a r e an t h e i r mss s h e l l and have a d e f i n i t e mass. y

gn(E9,)

an

We take

say i n L n G (&anlge of s c a l e m a n s j u s t a

change An o r i g i n of y ) F

C3. -

P u t t i n g t h i s a11 together i t naems that. i n a hadron c o l l i s i o n A

m y t h i n g i n t h e GM systera p l o t t i n g x

+B

-J.

P /P f o r x negative the d i s t r i b u t i o n of produces Z

The i d e a t h a t t h e rlglrt mrovers depend only on A and t h e l e f t Hlovers only li k e 8 is c a l l e d l i d t i n g ( I ,e. a s P suggested by C.N. Yaag e t a l , &ys, Rev,

; .69)

framntation.

188 2159

It: was

(1969) but a t t h e time i t

was supposed t h e e regions separated and had ao m m m i c a t i o n , but i n f a c t

there i s a s e a bemeen,

This s e a , hotrrever, we suppose is universal and

(although i t l o g i c a l l y could) we a s s m e i t c a r r i e s no information from t h e r i g h t t o l e f t region. kle have o d t t e d t h e d i f f r a c t i o n d i s s o c i a t i o n but i t is evident t h a t i f i t is added. i n a d e f i n i t e percentage t o the i n e l a s t f c w i l l not change our

conclur3ion,

However, the f r a c t i o n t h a t e l a s t i c s c a t t e r i n g is of the t o t a l

cross s e c t i o n does not Beem t o be universal ( f o r e x a q l e , f o r pp a t l a r g e P aeg/ctot

%

.25 whereas f o r 'n

p i t is c l o s e r t o . l 7 , s e e G. Giacomelli i n

Proceedings h t e r d a r n Conference an Bleaentary P a r t i c l e s , North Holland Press (2971)).

Thua lim%ting f r a e e n t a t i o n cannot be absolutely exact.

It

is probably generally nearly c o r r e c t ; perhaps i n a f u t u r e more exact under-

standing i t w i l l be t r u e f o r p a r t s of the c o l l i s i o n characterized by a o m other p a r m e t e r (e,g. impact, paratweter) b u t when i n t e g r a t e d over t h i s p a r a w t e r i t i s no longer exact f o r d i f f e r e n t cases give variocls d i f f e r e n t r e l a t i v e

weights t o t h e various values of the p a r a m t e c ,

Nevertheless even with t h i s

evidence a g a i n s t its p e r f e c t universal v a l i d i t y we continue to ancllyze i n a naive and s h p T e way leaving refinements t o some f u t u r e date, m e r e a r e a n m b e r a f a d d i t i o n a l conclusions m d e by assumning the b r k o v i a n fdea and extending our ideas such a s about gaps A8 from parton t o hadron wave functions.

M e s h a l l not discuas the= i n d e t a i l f o r hadron

c o f l i s f o n s i s n o t our caain s u b j e c t but give s o m e a w l e z ; .

We a s e m a ,

analogously t o A8 t h a t

(right

whiag

p a r t i c l e q u a n t m a m b e r s of A ncFnus quantum n d e r s of

a l l hadrons t o t h e ri&t of t h e gap), A+B

121

-c

For e x a w l e , f o r t h e exclusive c o l l i s i o n

C+D s o the outgoing s t a t e is pure C t a r i & r , B t o Ieft, the gap is

~ +fin2 ~ 1 PB/ 1

an4 1 P

~ = Pens ~and~ e-aby = ( 4 ~ ~ =~ B-'~ 1where ~ ' a depends

on t h e quantum nu&ers of A-C,

should go a s s-~('-')

Looked a t fuon a Regge point of view t h i s

whose a depends on the quantm. nunabera exchanged i n t h e

t channel, which is the @ama s A-C,

Thus a is i d e n t i f i e d with 2 f l - a ) ( o r

whatever the c o r r e c t pawer law of e n e r a f a l l - o f f turns o u t t o be) m d we mke a contact with t h e theory of exelusive r e a c t i o n s ,

(The s a w goes i f C is

two p a r t i c l e s l i k e n f p of fixed t o t a l mass2 n o t n e c e s s a r i l y a t resonance. X do not know of examples o f f resonance where t h i s pawer law has been checked

-

again we s e e the universal p r i n c i p l e t h a t going t o higher energy does not. l i f t a resonance ever higher against '"on-reamant'$

badground

- the

l a t t e r can

always a l s o be thought of a s taiLa of o t h e r resonanceo,) Again applying t h i s t o the case t h a t C only is near t h e end of its range, i o nearly 1 but D is anything, even m a y p a r t i c l e s , we s e e we a r e

so x

generating a gap of en(l-nc)

and the q l i t u d e goes a s e-aPn('-xc' a(l-xc )a-ldxc with a

The d i s t r i b u t i o n of xc is then d(l-xcla

Thie r e s u l t , though "'legally" t r u e a s P a nearly unobservable. collision. of rnllss from x

xc

-*

1,

2-2o.

is i n p r a c t i c a l eases

For example suppose C i s a proton produced i n a p+

Protons a l s o come from d i f f r a c t i o n d i s s o c i a t i o n of a reeonanee of

% say

-

m,

= (I-X~)~.

going t o proton W and pion.

(E -p )/$ t o x P P

-

and mmnturn of the proton

9;n

)/S

(as P + -1, where Ep, pp are the energy P the r e s t f r a m of the resonmce, Although t h e

(E +p P

This s p i l l s protans over a raage

l a t t e r is l e s s than one, t h e r e is a vary 811li311 gap (of range .98 t a l f o r %2

2.16) f r e e of d f f f r a c t i o n generated procons i t is too small t o i s o l a t e

experlentally.

Z f a gross p l o t is made f o r xc not s u f f i c i e n t l y near l

various nu&ers o f d i s s o c i a t i o n proton& a r e included and the v a r f a t i a n of numbers appaars f a r from ( l - ~ ~ 1 ' ' ~ ' d x ~ . This d i f f i c u l t y does not a r i s e f o r pions when t h e i n c i d e n t p a r t i c l e i s a proton,

Lecture 54

I should l i k e t o m k e a few coments about, our "conclu6sion" fraza C2, PSrst slnccz t h e m a n number of p a r t i c l e s goes as dxlx i n t h e smll region and continues across x

O a s dp/f

i t is evident t h a t the t o t a l n u d e r of particles

of a given kind ( t h e m u l t i p l i c i t y f o r t h a t kind) r i s e s logarithmlclal1y with fLnP o r with Ilns.

This i s a l s o obviow i n tha a r e a of the y pilot where the

plateau expands logarithmically with s.

But the plateau region is ( s t a t i s t i c a l l y )

n e u t r a l , f t s average f o r any a d d i t i v e qucsntm n u d e r such as charge, t h i r d component o f isospin, bargon nu&er, hypercharge, z cawonent o f angular

mmnturn e t c . , must be zero (because i f not

would give a Rns dependent

&/X

value fox one of these fixed conserved quantuna riders). t h e cascade i d e a of how the plateau is f a m e d , 1 -

dx (Number of a* a t x

- Number

of n- a t

X]

We expect t h i s from

Thus such i n t e g r a l s a s

conver*

t o nurabers which a r e

O c h a r a c t e r i s t i c of the p a r t i c l e A ( i n i t i a l l y taaving t o t h e r f g h t ) independent of P a s P

-t

=.

i n t e g r a l Ear

Independently carrespanding "&eft numbers" l i k e t h e s&m -1 t o O can be deflned, tJhich should iiepend on B.

X

I n p a r t i c u l a r then we. can d e f i n e d e f i n i t e q u a n t m nuoibers ( f o r the a d d i t i v e quantum n m b e r s ) f o r t h e right-rrroving particless3,by simply adding the t o t a l nuniber f o r a l l f o r kihich

X

0) i n t h e CM system.

O ( i . e . pZ

This nuaber w i l l vary from event t o event, of course, but we want t h e s t a t i s t i c a l

expected man over many events.

a constant a s s

+ we

This "mean r i @ t quanturn rider" w i l l approach

Thus we can t a l k of the "right mean 3-isospfn" o r the

"right m a n strangeness".

I t is evident t h a t these mean r i g h t q u a n t m n w b e r s

must, under t h e i d e a s of C2, be t h e same a s those of t h e i n c o d n g right-moving particle.

The plateau region does n o t l e t any qwntum nmbexs s l i p through i t .

( I f f o r a x m p l e we take a s p e t r i c c o l l i s i o n A nmbar conservation and must be t h a t of b. t o ntake A

+B

+A

then by o v e r a l l quantum

t r y the r i g h t q u a n t m a m b e r s (and t h e l e f t )

S

But by l i m i t i n g f r a ~ e a t a t i o nreplacing the l e f t A by B

does nog change t h e d i s t r i b u t i o n of r i g h t movers hence they

s t i l l carry t h e quantrna n u h e r s of A,) 'I'hus f n t e r e s t t n g l y a s P

+

the right-moving p a r t i c l e s t n the mean carry

t h e energy, t h e atowntuna ( d n w a constant), the 3 i s a s p i n , strangeness, baryon number, z angular w-mentum, e t c . , of t h e incoming r i g h t mover. NOB: -

We show t h a t , disregarding q m n t i t i e s of order 1/P, the difference of

t o t a l energy E and t o t a l ~ o m e n t wP of t h e p a r t i c l e e moving t o t h e ri+t is a constant D

.P

C(ci

- pZi)

(independent of P a s P

-t

"10

and i n f a c t i f t h e plateau For f i n i t e

is universal, t h e same constant D f o r every p a r t i c l e A).

positive) the difference from one hadranic p a r t i c l e i s c (p

2 2 .(m

A

X

(say

- Px

-p

)/2Px (m i s the m ~ of s the hadron) which is of order 1/P and t h e r e f o r e

negligible.

I"fie main contribution cornea from x near zero where the d i s t r i b u t i o n

of a p a r t i c u l a r type is edplc, hence the contribution t o

E

- p of

these is

Hadron-Hdron Collisions af Extreme Enargitls

c

j

( E - P ~ ) ~ P ~ / E~

.

m

The i n t e g r a l gives

so D

C

P ,"""Q

i n the plateau.

I f t h e p l a t e a w a r e universal, the constant II i s universal

and m y be e a s i l y calculated i n t e r n of already nreasured quzmtities, Xbua i n a c o l l i s i o n of 8 and B each p a r t i c l e i s converted i n t o a t r a i n of p a r t i c l e s rnoving i n i t s own d i r e c t i o n .

The " t r a i n A"

the q u a a t m numbers

"S

of t h e p a r t i c l e A and its energy (by the cansertvation of energy) but has l o s t a c e r t a i n m o e n t m D i n the i n t e r a c t f a n , i t i r s held back a b i t by the i n t e r (Such a f i n i t e moment= t r a n s f e r

a c t i o n , A and B each l o s e D t o the o t h e r ,

is, of course, c o n s i s t e n t and understandable i f only wees i n t e r a c t i n the collision.) (For the wave function of a s i n g l e hadron described i n a s s m p r i o n s

A2 t o A6, t h e t o t a l aomentunt of t h e partons is, of course P, the t o t a l ntomentm of t h e s t a t e , but t h e t o t a l energy energy E

is not the t o t a l

C i

P because of i n t e r a c t i o n energy which compensates t h e expected

f l n i t e excess of Cci above CpzI.) As a f i r e t s t e p t o describe these things f o m a l l y , we a r e t r y i n g t o describe t h e s t a t e /Ain right' Bin left' ( h e r e "h riight" Beans having very l a r g e p o s i t i v e Iongftudinal momntum: P, and "in l e f t " m a a s -P) i n t e r m of autgaing hadron s t a t e s the a w t r f x ,

- an e l e w n t

of

O f course the mst l i k e l y thing is t h a t the two p a r t i c l e s

do n o t c o l l i d e , making simply

/aut

Bout

left>.

Ve wish t o d e a l with

t h e wave functions i f they c o l l i d e , s o we w r i t e as usual S a r e speaking of the T la;atrk, o u t l i n e our ideas.

l

+ iT

and we

kfe s h a l l not n o m l i z e i t c o r r e c t l y , but j u s t

F o m l l y t h i s wave function can be given i n t e r m of the

amplitude t o f i n d various outgoing hadrons.

I f c* is the (formal) operator

t o represent the c r e a t i o n a f so@ kind of a hadron (kind, transverse momntm pk, and l ~ n g l t u d i n a lmowntum p a r e i n d i c e s of c*) we can represent such s t a t e s by

XIVAC,where

d i s c w a i n g how X looks,

X is some operator function of the c*.

We have been

let M be an operator t o c r e a t e a p l a t e a u

r r p i c a l u a i v e r s a l plateau f o r s o w range of x around O

- say x

- say

a

-.2 t o +.2

Photon-HadronInteractions [ t h e exact way the plateau of M c u t s off f o r f i n i t e x i s arbitrary; its choice a f f e c t s the d e f i n i t i o n of G ~ , Gdefined ~ l a t e r , but the f i n a l operator Next we w r i t e X a s G ~ G % where G R is t o mrrdify

X i s n o t dependent on t h i s ] .

the s e a on the r i g h t ( f o r x r 0. X t involves c r e a t i o n operators c

*

t o add

p a r t i c l e s t o (and beyond) t h e plateau operator M, and m a i h i l b t i o n o p e r a t o r s c t o take p a r t i c l e s o u t (which were put i n by our a r b i t r a r y choice 05 how the 3 O ( t h a t i s the meaning af the R). L * Likewise G is an operator function of c , c s n l y f o r x < Q. The operators L R

plateau M i s defined) but a l l these f o r x

G ,G c o m u t e s i n c e they contain operators of d i f f e r e n t p a r t i c l e s (soate s i g n s must be adjusted f o r Fermi p a r t i c l e s ) , R,

L)

Thus we w r i t e

G ~ G % IVAC, A B

where we have w r i t t e n [M-plateau3 = H/VAC>. on the p a r t i c l e A, e t c .

(54.3)

R

The operator GZ depends only

I f you want things t o look even n i c e r w r i t e the l e f t

s i d e i n terms of operators too, say d* which c r e a t e incantag p a r t i c l e s , and then have

R*% L* VAC, = ~ , " ~ i / ~ - ~ l a r e a u ,

TdA

* is equivalent ( i n t h i s two-body equation a t l e a s t ) t o R* B but i n an odd representation i n which dA a c t s on the vacuum and CA on

Thus t h e operator d: G:,

the i4-plateau s t a t e . A rasearch problem which f s very intportant, and v i r t u a l l y u n k n m

t h e o r e t i c a l l y , is ( t h e very r a r e ) c o l l i s i o n s a t e x t r e w e n e r m i n which the r e l a t i v e moaenta t o t h e o r i g i n a l d i r e c t f o n .

p a r t i c l e s come out a t

Fox:

example, proton-protm e l a s t l c s c a t t e r i n g a t f i n i t e angle, e.g. go0, where t i s the same order a s s a s

S

-r"

m.

What kind of physical view accounts f o r these

c o l l i s i o n e , I s h a l l not discuss ideas which have been t r i e d here, f o r our s u b j e c t is photons, but s h a l l only Coment t h a t nothing is c l e a r l y understood and you can s t a r t from s c r a t c h on your own. have t o be abandoned o r quantified?)

(For example w i l l assumption B1

(You s t a r t by looking f i r s t roughly a t

the experimental r e s u l t s t o r e m & e r q u a l i t a t i v e s a l i e n t f e a t u r e s t h a t d g h t need explanation,) nts:

P

By assuming t h a t the wee region is t h e same f o r each hadron and t h a t

Hadron-Hdmn Collisionsat Extreme Energies only vees i n t e r a c t have we not a s s m e d t h a t a l l t o t a l c r o s s s e c t i o n s a or a PP =P e t c . a r e equal, c l e a r l y contrary t o f a c t ? 1 have n o t thought t h i s out c l e a r l y but have always supposed t h a t t h e p a r t of t h e wave function which does i n t e r a c t (which i s always i n f i n i t e s i m a l conrpared t o t h e p a r t where they go past each o t h e r without i n t e r a c t h g ) could s t i l l have soare a o m l b z a t i o n r e l a t e d t o the t o t a l c r o s s s e c t i o n f o r t h a t p a r t i c u l a r c o l f i s i o n without being inconsistent: with o t h e r ideas. each G

A

Ta the forrnal expression above, f o r example,

could carry a numerical c o e f f i c i e n t gA proper t c A. 2 2

t o t a l c r o s s s e c t i o n s proportional t o gAgg,

00

'This would ntalse

a s is s a i d , f a e r a r f z a b l e ,

It

m y be. t h a t the previous a s s m p t i o n s do n o t i a p l y t h a t the totall c r o s s s e c t i o n s

a r e n e c e s s a r i l y equal, but r a t h e r perhaps t h a t they a r e f a c t o r i z a b l e , imply f o r example, t h a t a

- \/I;-

/b

PP \ ns

etc.

I C would

W e do not have any evidence on

whether t h i s is t r u e .

Tn these s t u d i e s we have made no r e m r k s which p e m i t us t o understand transverse =ownturn behavior (except t o say t h a t transverse nrromenta i a hadronic c o l l i s i o n s a r e limited, a r e s u l t taken d i r e c t l y from rrxperirnent].

Obviously

l o t s of i n t e r e s t i n g t h e o r e t i c a l questions r e m i n , such a s what functfon is t h e trcinsverw mmntuar d i s t r i b u t i o n , how does i t d i f f e r f o r various values of o r f o r n%

arid K's?

X,

Now should exclusive cross sectiona vary with t , e t c . ?

This e n t i r e realm of phenomena has been l e f t out of our a n a l y s i s , an e x c e l l e n t f u t u r e opportunity f o r advance l i e s here,

Hadronic States in Deep

astic Scattering

Lecture 55

We a s s u m t h a t i n the o r i g i n a l f i e l d Ea<oniain describing hadrons i n t;em of partons t h e r e a r e t e m givlrrg t h e coupling of partons wllth the

vector p o t e n t i a l of t h e quclntunn e l e c t r o w g n e t i c f i e l d .

We s h a l l assuvle i n

the s p i r i t of .olinIaum e l e c t r o ~ g n e t i ccoupling t h a t they couple i n t h e sixnplest m y expected Eram t h e propagation operators v i a gauge inrrariance.

That is we

IBSUIEIB:

That is a l ~ at h e coupling t h a t would be vtllid i f they were i d e a l f r e e p a r t i c l e s , This coupling Is not unique i f t h e partons a r e spin. 2 o r higher, but f o r t h e present t h i s w i l l not coneem us f o r we s h a l l suppose partons a r e e i t h e r s p i n O o r s p i n 112,

AZthou*

we a r e i n danger of not having t h e moat

general caae we s h a l l nevertheless explicitly next take the working hypothesis (suggested, of course, a s we have discussed by experixaent on vWZ and W1 and not a p r i o r i by theory) t h a t

(a i s an index f o r the kind of partan).

250

251

Final Hadronic Smrm in Deep l ~ e l ~ z sSc~tterr'ng ti~ We have seen how t h e assumptions AP-A8 plus these two D1 m d D2 l e a d t o the s c a l i n g expectations f o r t h e deep i n e l a s t i c seattergng ( l e c t u r e 27) m d Urnever, here w e s h a l l

there i s no reason t o repeat a l l t h a t h e r e again.

discuss what we can say about the products i n photon c o l l i s i o n s , i n p a r t i c u l a r we begin with the deep i n e l a s t i c ep s c a t t e r i n g regian cg2

-2Mvx.

Mv

tm of the proton, q t h a t of the v i r t w l photon) s o

P*q

virtual

V

photon energy i n laboratory (proton at r e s t ) system* kt us use the coordinate

-

system wfth t h e v i r t u a l photon purely spacelike qufi (0, -ZPx,O,O) PMm(P,P,O,O) 2 92 4 p 2 x 2 , 2 Hv 4P x. Then a s a r e s u l t of our avsvmptions t h e parton wave function before and a f t e r the c o l l i s i o n s looks l i k e :

Imdiately

A f tl?r

&up l i n g

That Is, one parton (say type a) moving t o the l e f t , the r e m i n i n g partons mvin,ng t o r i g h t j u s t a s i n o r i g i n a l proton, l e s s the individual parton a of mmentum x.

The r e l a t i v e p r o b a b i l i t y of t h i s p i c t u r e is e t n a ( x )

is t h e nunber of partons of type cr wSth pz/P

fi

X

where na(x)

i n the origilzsl proton s t a t e .

Then:

.

The t o t a l cross s e c t i o n i n t h i s sca3e

is a superp o s i t i o n of cases of d i f f e r e n t typee a of partons with weights,

on t h e character of the c o l l i s i o n through x ;

wa(x)

kla depending

2

e;na(x)/T

eg ng(x) ; l3

the s m of the weights being l.

h e obvious consequence of t h i ~and our other a s s m p t i o n s is t h a t i n t h i s system

{for fixed x a s we very F,

o r i f you l i k e v). The longitudinal m o a n t a w i l l s c a l e as P, %.e, i f they a r e s t a t e d i n

u n i t s of P a s say 1-tP t h e d i s t r i b u t i o n s w i l l be independent of P a s P (depend only on

Q),

We expect aZso near q equal zero t o find a dq/n behavior, Q

+

For p o s i t i v e

(night movers) we expect. i t behaves likrtt t h e universal H-plateau charaetlerirstic

of a wave function l i k e B3, which we know from hadron colZisione,

Pox negative

1-t wcs axe i n the ""plateau regian of i n i t i a l parton gap" Mefined i n connectfan

252

Photon-Hadron Interactions

with Cl),

We have not assmecl these two p l a t e a w a r e t h e s a w s o t h e c o e f f i c i e n t

of dr1ll-r need n o t be the same.

1 E they a r e not we s h a l l have trollible defining

what happens i n t h e t r m s i t i o n regian

- i t cannot

Leads t o the same c o e f f i c i e n t f o r plus and minus

go simply a s Cipz/c f o r t h a t

n. We s e e however t h a t t h i s

question is t o some e x t e n t an a r t i f a c t of aur p a r t i c u l a r choice of caardinate system.

Note t h a t the s t a t e described here a s "immdiatelp a f t e r coupling" i s

an i n i t i a l parton s t a t e ( i n the sense discussed i n t h e xemrk following the discussion on Cl)

- there

s t i l l must be i n t e r a c t i o n s from t h e Hamiltonian

before i t becomes the '%appropriatem outgoing wave function.

%is w i l l produce

cascading a f t h e left-moving parton i n t o t h e gap smearing the wee region i n t o negative

Q

and making l a r g e readjustments f o r the r i & t - m v i n g

(because, f o r f i n i t e

X,

system a l s o

they a r e no longer t h e c o r r e c t s o l u t i o n of W$ = EJf

s i n c e one parton is missing), This a l l appears q u i t e complicated and i t is d i f f i c u l t t o make firm predictions,

However, we might cant inue to assume i n t e r a c t i o n s a r e each Limited

i n range on a r a p i d i t y p l o t I n gaps, e t c .

- although t h e r e

a r e m n y of them possible f i l l i n g

But we s h a l l t r y t o adhere t o t h e grincllple a t l e a s t t h a t the

parton mavfng t o the l e f t

d e t e m i n e a t h e f i n a l hadrons t o t h e l e f t

and lgke-

wise f o r t h e r i g h t . We put t h i s idea f o m l l y i n t o t h e following a s s m p t i o n ,

a g e n e r a l i z a t i o n of Cl, 62 (we w r i t e i t independently, f o r i t may n o t be t r u e while t h e s p c i a l case Cl o r C2 m y be).

C5. I n the c e n t e r of massi system ( o r one m v i n g l o n g i t u d i n a l l y a t m y v e l o c i t y not near c)

(Lfkewise , exchanging l e f t and r i g h t . ) Assumption CS, i f i t were r i g h t and t h i s c o n t i n u i t y i n iLndPZ/c, would seem t o suggest t h a t both plar;eatts f i t tagether.

I arn not s u r e of myself

h e r e but s h a l l put i t down as an e x p l i c i t a s s m p t i o n which would, i f i t i s t r u e , r e w v e a l l our d i f f i c u l t i e s t h a t of hadron-hadxon c o l l i s i o n s .

- the

dtl/rl

region i s always universal,

233

Final Hadronic States in Deep Ittelastic Scatrcrifrg

This a s m t i o a i s a t present, very weakly based and m y e a s i l y be wrong

- it i s an i n t e r e s t i n g conjecture,

We now make a more d e t a i l e d discussion of our expectations f o r the l e f t -

moving p a r t i c l e s , (1 have p r o f i red g r e a t l y from conversations with A. Cisneros on these m t t e r s , )

For these p a r t i c l e s a v a r i a b l e more convenient than rt (which

goes d m t o -x) i s z

- rl/x

-p,/Px

= P*p/Psq the f ractton that the l e f t Since t h i s

nnovhg p a r t i c l e ' s moraenturln i s of the t o t a l left-moving nroaentuta.

is z = P.p/Peq i t is t h e energy of t h e p a r t i c l e i n t e r n of t h e energy v of t h e photon i n the laboratory system, how t h e v i r t u a l photon f r a g m n c s ,

X t is t h e proper v a r i a b l e f o r seeing

Of course a@ v

4 m,

x fixed, t h e d i s t r i b u t i o n s

i n z s c a l e a s we have s a i d ,

If we could be s u r e t h a t only a partoa of type a c m e out (which by t h e

way, can be much more n e a r l y done f a r neutrZno s c a t t e r i n g

- i n the quark m d e l ,

neutrino scattering can l e a d t o unique quarks t o t h e l e f t ) t h e d i s t r i b u t i o n t o the l e f t would be unique

- say

a function of r only D : ( , )

the p r o b a b i l i t y of n y K a t zl,z2 depends on

z l ~ Z as

7ite

Da (z 1"z )

- and

Dty(z).

"fhe n a b e r of v % with a given z i s

these functions do i n no wily depend on

X.

m e y do not

depend on x because the hadrons t o t h e l e f t d e ~ e n donly on the parton t o t h e l e f t ( a ) axld the adjacent wees from t h e hadron ( i f a t a l l )

- and

these

l a t t e r a r e m i v e r s a l and unaffected by the ternoval of t h e parton a t x from the proton,

The l a t t e r does not a f f e c t l e f t - m v i n g h a d r o n ~ f o r its r e l a t i v e

mmntrura t o l e f t - m v i n g hadrons i s not f l n i t e but g r w s a s P

4 0,

TIze a c t u a l d i s t r i b u t i o n seen a t a given x w i l l depend on x because the r e l a t i v e p r o b a b f l i t i e s of produchtng d i f f e r e n t kinds of partons or w i l l depend on x,

The a c t u a l dSstributions DD(x,z) w i l l be the weighted average f o r each

where the weights wa(x) proportional here t o eeLna(x) a r e defined by

These functions D (z), o r equivalently t h e i r c r e a t i o n operator

R on

K-plateau ( i f 65) isolates sontethinlg charaetaristic: of gartons and, i f our

254

Photon-Hadron Interactions

a s s w t i o n s a r e a l l c o r r e c t , therefore very fundamntal. indeed,

Me s h a l l

discuas l a t e r a s p e c i f i c parton -&l (quarks) a s well a s s o w p r a c t i c a l questions about t h e possible e x t r a c t i o n of t h e i s o l a t e d R (z) f r o a experilaent, a s well a s t h e p o s s i b i l i t i e s of finding t h e u

D

may behave,

(2)

(X)

by s p e c i a l guesses about hou

To raa t h e p o s s i b i l i t y of s p e c i a l f u n c t i o m c h a r a c t e r i s t i c

of each kind of parton i s a very i n t e r e s t i n g p o s e i b i l f t y , and one t h a t could be an entrance t o a path i n t o t h e h e a r t of t h e m c h a n l s w of strong i n t e r actims, These same functions D

(2)

- for

w i l l appear i n c e r t a i n o t h e r experilsents

e x a m l e , of course, i n deep neutrino proton

i.

p

+ products

experimnts.

The

a n a l y s i s is nearly t h e saae a s h e r e except t h a t t h e f m d m n t a l coupling m y be d i f f e r e n t so although n Agein i n the e*e-

(X)

a r e the s a a the weights wo(x) c o w o u t d i f f e r e n t l y ,

collision the a s s q t i m D2 says our i n i t i a l s t a t e

is j u s t a p a i r , parton o and a n t i p a r t o n

2 R L C6 t h e f i n a l s t a t e would be C ea (Da D;

with weight eo2,

+ :Da

a

Thus, ass&ng

/ ~ - ~ l . a t e a uagain > producing

hadrozls i n any one d i r e c t i o n characterized by the dlstributioln

where we sum an a over parrcns and a n t i p a r t o n s , I f , f o r e x a w l e , i n a o m experio~sntwe could be s u r e t h a t a c e r t a i n parton a erne o u t t o t h e l e f t say, then a s we have seen we would expect t h a t t h e t o t a l "leeft-mvbg quantrrrn rider" "he

sm, f o r so= a d d i t i v e quasltm

a m b e r of t h a t n d e r f o r a l l hadrons arrving t o t h e l e f t averaged over a l l events) w u l d be t h a t of the parton a ,

Thus i n p r i n c i p l e we could define

o r d e t e d n e I n tern of e ~ p e r i m n tt h e q u a n t m n m b e r s of the partons, the s t a t e i s not pure we s h a l l have t o Imm something of: the weights d

e t h i s useful

- but there are s o

\(X)

If to

different: kinds of e x p e r i w n t s

possible t h a t i n p r i n c i p l e the wa(x) can be d e t e m i n e d a s well. a s t h e o v e r a l l q m n t m a m b e r s of the partons, The p a r t i c l e s t o t h e r i g h t u n d e r h e p i n e l a s t i c e-p s c a t t e r i n g corn f r o n Erawentation of a proton with one cr parton of wroentw f r a c t l o n say E (p_a,x) ( E ) .

They a r e evidently not very fundamental.

X

remved,

But i t is c l e a r

t h a t the sam kind of f h a Z s t a t e r e a u l t a (on both s i d e s , l e f t and r i @ t ) i n Rrell"

experiment p

+p

J-

p

+ + 1-1- + any

hadrons s o t h e products i n t h i s

255

Final Hadrank States in Deep Inelastic Scattering e x p e r i m n t can be e n t i r e l y expressed i n t e r n of these E (p-cr,x)

and hence

has been worked out) i n t e r n of the products for deep

(supposing n,(x) ep s c a t t e r i n g .

"

We

leave i t f a r you t o w r i t e t h e e x p l i c i t r e l a t i o n s and t o

suggest p r a c t i c a l experfmn ts t o t e s t your ideas.

Our f i n a l hadron s t a t e is according t o our assunptions

NOTE:

L

V,'"' 4,

R

1%

(p-~,3

plateau,

,.I i s t h e operator f o r a l e f t partan a , and EB

where B

moving f r a g m n t s .

for the right(p-a,x> But t h i s can be considered a s a lanemnic only f o r the

expression i s probably ilnpossible f o r one operator M allways. would seem t o prevent us from w r i t i n g quanturo n u a e r s of two quarks (or

+ B)

a

Cz

plateau, which would have t o t a l

whi& is impossible t a w r i t e i n termss

of the hadron operators having m l y i n t e g r a l quantufo nmbers. to J, Wndula f o r pointing t h i s o u t , )

Because nothing

(1 a a indebted

A v a l i d m t h e m a t i c a l representation

f o r these ideas is an e x c e l l e n t problem. The reader shauld be warned t h a t a n d e r of these s c a l i n g predictions

f o r s p e c i a l produets of r e a c t i o n s may only hold a t much higher energies than t h a t a t which s c a l i n g f o r t h e t o t a l cross s e c t i o n (vW2 and Ml) s e t s I n , This warning r e s u l t s from t h e o r e t i c a l experience with a n m b e r of e x a w l e s of a n a l o g ~ u st h e o r e m i n n o n - r e l a t i v i s t i c quantum nogchanis where the sum works we11 before the i n d i v i d u a l t e r n do,

This is becautlre i f c e r t a i n i n t e r -

a c t i o n s a r e disregarded I n working t h e t o t a l p r o b a b i l i t y by assuming c e r t a i n s t a t e s only a r e "entered" subsequent gnteractions naay not change the t o t a l p r o b a b i l i t y t h e s t a t e was ' % ~ ? ~ e r e db"u t atay r e d i s t r i b u t e t h a t p r o b a b i l i t y over d i f f e r e n t f i n a l s t a t e s than were expected,

In t h e s p e c i a l case of s w f l

X

t h e predictions a r e e s p e c i a l l y sitaple,

F i r s t consider t h e r i g h t s i d e ( o r i g i n a l proton).

Here we? have a parton d-is-

t r l b u t i o n j u s t l i k e t h a t of a proton with only a very low x parton rernoved aad the wees disturbed (by i n t e r a c t i o n with t h e plateau developing from the left). a proton

Thus a l l t h e partons o r any s u b s t a n t i a l x a r e exactly l i k e t h a t of

- and we

(at least for z

can expect the saate d i s t r i b u t i o n of hadrms t o c o w out X)

a s do coae out f o r a hadron c o l l i s i o n of a proton, say

or $(E).

Hence f o r n small ER

Jil

(p-a,x> p' Next f o r amall x a l l na(x)dn go a s Ca &/X where Ca is a constant, ~o

t h a t wafx)

small

C,/):

CB

m

y

approaches a constaat y

B

Next call. llL t h e mixture :D

X.

r

parton, each wei@&witb f o r smll

X,

weight ya.

-

independent of x f o r

z ~ ~ D o: f t h e d i s t r i b u t i o n s f o r each

Our hadron d i s t r t b u t i o a becoma thus,

nearly

That is, f o r small m t h e proton f x a p e n t e i n r o a f o r a independent of x and the ss=

a s i t does f o r a hadron c o l l i s i o n .

E r a p n t a i n a unfversal way independent of X

Bnd the v i r t u a l photon a l s o X.

Sfnce we have afisawd t h e low

region the sam f o r a l l hadrons, t h e Col and yap and heace By do not depend

on the p a r t i c l e s t r u c k by t h e photon f n o m l i z e d t o t h e t o t a l cross-section f o r c o l l f s i a n , of course).

A sntall,

X

photon and

rt

hsdron behave j u s t l i k e the

c o l l i s f o n of two hadrons, each f r s g m n t s i n its m c h a r a c t e r i s t i c way,

That

of t h e photon f s independent of X,

2 For f t n i t e q, negative q we can s t i l l use our system of coordinates i n rJhtch q has only a space component Q.

exrzept i f Q

0.

It fs c l e a r h e r e hor~evelrt h a t only x near zero can be

a f f e c t e d by t h e *ocon

They a r e ,

Q; t h a t i s only t h e wees a r e e f f e c t e d .

however, a f f e c t e d i n a very conrplicated way f o r i n t e r a c t i o n i s important i n t h e wee region.

%a cannot t h e r e f o r e p r e d i c t whet vi3.3, happen t h e r e , but we

can note (a) t h a t i t is the s m f o r every hadroll A, A

+y

-c

products f o r

we have the @amweea f o r every hadron accordLng t o A&, and (b) t h e fragm n t a t i o n of the f i n i t e m i n t h e above system is c h a r a c t e r i s t i c of partons of system A only, f o r only t h e wees a r e e f f e c t e d by t h e photon.

In consequence of (a) the products on t h e l e f t rJhillh can be Ctczscribed

2 &ere Peq

i n t e r n of z

P Is t h e proton f o u r + ~ n t u n r , p is t h a t of a

product and q t h a t of the photon, f o r f i n i t e z m v distribution D

Y19

+

=, is so= kind of a

Z ( ~ ) . The d i s t r i b u t i o n c l e a r l y depends on

the v i r t u a l

257

Fhail Hadronic States in Dwp fnelastie Scatterkg QE the photon, b e e a w e the c o q l i c a t e d i n t e r a c t i o n s of the wees depend

=ss

on t h i s nrawntusn.

I n the o t h e r d i r e c t i o n ( t h e v a r i a b l e (q*p/q*P) the proton

f r a p e n t s i n t h e B a E way a s i t does f o r hadron c o l l i s i o n s , tions hold f o r q2

These eonsidera-

O a l s o , of course, but our coordinate system is inconvenient

f a r such a c a s e , kfe could a l s o use the c e n t e r of =ss

system f o r any fin.nite q

Conservation of energy and momentm meaQs t h e v i r t u a l photon (P'

2

+ m,qZ

finite)

i n t e r a c t s only wlth t h e wee partons of t h e t a r g e t proton ( o r hadran G).

This

i n t e r a c t i o n i s conrplicated but produces t h e s a w d i s t r i b u t i o n f o r any hadron f o r given q

2

.

The h a d r m behaves a s i t always does where

turbed whether by another hadron o r by a photon,

i t s wees a r e dis-

P o m a l l y our f i n a l hadran

s t a t e is

%us a s f a r a s h i @ energy i n e l a s t i c c o l l i s i o n e a r e concerned t h e (virtuill o r r e a l ) photon s e t s j u s t l i k e a hadron inasmuch as It appears t o have

itcl

2

2

own (q dependent, o r q 4 1 £ r a m a t a t i o n products, i n its;

d i r e c t i o n , the hadron f r a p e n t i n g a l a o i n Its c h a r a c t e t i s t l c way, IPhis af course makes a n i c e union with the- i d e a of vector =son

dodnance,

t h a t a f r e e photon (q2-0) has a c e r t a i n reasonable p r o b a b i l i t y t o be a v i r t u a l vector meson and a s such would behave i n hadrotl c o l l i s i o n s l i k e a hadron, note now

W@

We

s h a l l n o t have t o d e t e d n e with what prabab2lity i t looks l i k e

a badran and how t h i s varieJLee with q 2 , f o r i n any event i t , a s a whole, should a r t j u s t l i k e a hadron does i n v

In t h e c e n t e r of a r e te-

-S

+

-,a"

~olli~~eeion~r

p i c t u r e (and a l s o kn t h e spacelike q f i p r e ) t h e r e

of coupling i n which t h e p h ~ t o nE f r e t divides i n t o partons on the

way %n3 f o r e x a m l e one f a s t one slow, and these slow partons i n t e r a c t o r lsaxnihilate with t h e wee partons of the hadron.

Thus t h e p i c t u r e t h a t the

inco&ng photon loofts with some m p l f r u d e l i k e partons i t s e l f i s r e i n s t a t e d . lLsi g

2

r i s e s (and c e r t a i n l y where

X

2

= -q 12Mv is f i n i t e ) t h e contribution sf

aueh d2agram £falls away and only t h e d i r e c t coupllng t e r n of photon s c a t t e r i n g

a parton of the hadran reraain important.

F i n a l l y we a h a l l mateh our f i n i t e qZ region t o our smll we have done before we s h a l l suppose when

2

-

V +

.D

region.

ks

v is very l a r g e and -q2 l a r g e but

-q / ~ H vsmall the 1 i ~m ty be r a b n i n e i t h e r order:

r e s u l t e i t h e r from our f i n i t e q2, v

X

- i.e.,

formula o r from

X

we can get t h e

f i n i t e , but small,

fomula. mus (55.1) mwt agree with (55.2) f o r Large q

2

.

%is i s

easily done,

t h e r e s u l t s agree i f only we add t h e r e s u l t : DYsq2 = Dr f o r l a r g e g

2

.

mat

is:

2

The f r a p n t a t i o n produces of a photon of l a r g e -q becam independent of

2 2 -q a s -q r i s e s ,

(We a r e i n a l l czzsaaj, n o m l i z i a g t o the t o t a l cross s e c t i o n

which is varying, as l / q 2 , of course.)

Partons as Quarks

Lecture 56

We could now go on t o discuss various ntadels of what quantm n u d e r s partons carry, but we s h a l l l i m i t ourselves t o one e x a w l e , the one t h a t i s aost interesting,

The student should t r y other examples, suck a s the Sakata

nradef, t o see whether we t a n eliminate them by experivrents now done, or proposed, We s h a l l suppose the charged partom c m @ i n s i x v a r i e t i e s , three plus their antiparticles,

The three called u,d,s carry the quantum nu&ere of the

three quark ~ t a t e s(of the lw energy quark m d e l ) .

El..

This we s u m a r i z e by:

, Most o f our previous a s s w p t i o n s

were guided, o r we thought they were guided by f i e l d theory o r considerations based on hi& energy experimnts, guess,

Tkfs, of course, is not, i t i s an inspired

But tt i s a l s o contrary t o what can be t r u e i f the f i e l d ttreol-y i s too

ordinary,

For i a such a theory there would be a b m e s t a r e of quark nmber

me (and non-integral charge i n a localized wave packet) and, i n view of the cansemation of quark n u d e r , sone eigenstate of the system of quark nwaber one is expected.

I n other words we expect t o see r e a l p a r t i c l e s with quark

quantum n u a e r s .

They have not been seen.

they have very large =ss

- but t h i s makes

It is possible t o imgin~1:t h a t

i t very bard t o take a l l the

259

previous a s s m p t i o n s about a l l parton i n t e r a c t i o n s l i m i t e d t o t h e GeV region, etc.

There may be some way

t o reconcile a l l tlnis

intriguing; of t h e o r e t i c a l problms.

- i t is one of

t h e moat

I n order t o emphasize i t 1 w i l l make

another unneteeaary assumption t h a t I w i l l n o t use but I add t o remind you of t h e problem,

-

E2.

I f you p r e f e r t o replace i t by

"'physical quarks have high mass", go ahead

- you s t i l l have t h e o r e t i c a l work

t o do reconciling i t with E l and t h e r e s t of our a a s m p t i o n s , slay be wrong

- one of

Of c m r s e E l

the most important e x p e r i m n t a l jobs of the f u t u r e i s

t o f i n d out whether E l is indeed c o r r e c t o r impossgble and s o we should work o u t a s rnany t e s t a b l e consequences of i t a s p o s s i b l e ,

So f a r we have, a s

discussed i n l e c t u r e 32, corn t o t h e conclusion from expertmeat t h a t i n any ease :

.

E3,

%at they may be l i k e we do not now

know, except perhaps they m y not be vector (because the @ , p degeneracy i s not l i f t e d )

.

Although t h e problem of reconciling E l , E2 and f f e l d theory sl?igPtt be very d i f f i c u l t , i t appears ( a t l e a s e a t f i r s t s i g h t ) t o be n o t a t a l l d i f f i c u l t t o reconcile El, E2 and t h e o t h e r a s s m p t i o n s we have e x p l i c i t l y aade about parton and hadron d f a t r i b u t i o n s ! A c a r e f u l review of our assumptions shows t h i s ,

as already mentioned, i n

There i s possible doubt,

B1 ( i n t e r a c t i o n only between partons of small

r e l a t i v e y) but t h i s i s only used t o m k e the l a t e r asstznrptions = r e

plausible

and nraybe replaced by t h e s e l a t t e r aaswptfonap, A place where t h e r e Is an e s p e c i a l l y i n t e r e s t i n g , but n o t I n c o n s i s t e n t ,

conclusion is i n connectlon with '%ight+ov%ng q u a n t m nmbars" "iscussed

a

i n l e c t u r e 54) f o r the parton f a c t f a n Da and which should aow be non-integral, These n m b e r s a r e defined s t a t i s t i c a l l y a s t h e average over a l l events

-

although each event must give i n t e g r a l values t h e average, of course, need n o t be,

For e x a m l e i f we know one quark (and no antiquark) is s e n t t a t h e f f g h t ,

the mean n u d e r of basyon l e s s antibaxyons found on the r i g h t should (at extremely high e n e r m , a t Least) be

+

113.

The a r g u e n r s leadizleg t o t h i s concfusion a r e not obviated by the f a c t t h a t t h e partons do not have i n t e g r a l quantunr a m b e r s .

Xmgine f o r a very

Parterrs as Quarks l a r g e %nP t h a t the quarks are d i s t r i b u t e d i n the ( f h a l s t a t e ) wave function i n a long plateau:

( t h e l e f t end of the plateau is generated by a long chain

*

il:

of cascades v i a t e r n s l i k e a a a from t h e i n i t i a l s i n g l e pubirk)

Then i n turning i n t o f i n a l hadrons various bundles of quarks go together t o make l e g i t i m a t e hadron quantum numbers.

I n doing so they take combinations

of quarks over a f i n i t e range o f y a s i l l u s t r a t e d above.

(The o v e r a l l t o t a l

t r i a l i t y must be zero, of course, f o r the i n i t i a l s t a t e has zero t r i a l i t y ; our i l l u s t r a t i o n t h e i n i t i a l s t a t e has baryon n u d e r one,)

in

Me assume t h a t

t h e r e i s s o w non-zero p r o b a b i l i t y p e r dy t o pick of 3 quarks ( o r 3 antiquarks1 t o m k e a baryon ( o r antiibaqyon). t o pick up s t r a n g e quarks.

Likewise t h e r e should be a f i n i t e p r o b a b i l i t y

Pt is then seea t h a t a s a s t a t i s t i c a l matter with

a s u f f i c i e n t l y Long plateau kn y ((sufficiently long Wrkovian chain) t h e quark number (and strangeness) becows randomized m d the c e n t r a l region of t h e plateau is n e u t r a l oa t h e average i n these v a r i & l e s ,

This means t h a t the

r i g h t and l e f t mean q u a n t m n u d e r s of t h e f i n a l kadrons approach a constant t h a t depends a s we have supposed on t h e i n i t i a l r i g h t o r l e f t quark character. Pt doesn't m k e any difference exactly where you c u t t h e p l a t e a u i n deciding which is r i g h t o r which l e f t , a s long a s yclu c u t sowwhere n w r t h e ntlddle.

%is result: is s o i n t e r e s t i n g and Its e x p e r i m n t a l v e r i f i c a t i o n would represent such a d i r e c t measure of the supposed non-lntegral q u w t m numbers of t h e parton quarks t h a t we should say s o w wards about its possible v e t i fication,

F i r s t i n e l e c t r o n production i n general we do n o t have a s i n g l e

type of quark t h r m t o the l e f t ea t h e beauty of t h e result: would be confuged by having t o f i r s t know t h e wa(xl by ane e t h o d o r m o t h e r (see m t h o d s below). On the other hand f o r x near I we have been l e d by e x p e r i m n t ( r a t i o of

vWZn t o vWZp, l e c t u r e 31) t o suppose t h a t only t h a t is t r u e , near x

U

quarks survive

- hence i f

1 our left-moving quark may be a pure u quark.

h o t h e r way t o insure pure quarlcs is by neutrino s c a t t e r i n g which we discuss below, Secondly, although i t m y be easy t o f o m pions i n the plateau region

i t m y be harder t o f o m R's and s t i l l =re of t h e i r m s s e a ? ) ,

d i f f i c u l t t o f o r a baqona (becawe

If: t h i s is t h e case we should need a very lmng plateau

indeed t o g e t equilibriuan i n baryon a d e r , althou& hyperon charge a g h t be e a s i e r aad i s o s p i n e a s i e s t of a l l .

I n the e n e r m r m g e s a v a i l a b l e t o experi-

a n t therefore I would expect t h e quantum n d e r r u l e s t o work b e s t f o r iaospin, next f o r hypereharge, and b a v o n n h e r last ( i .e. requiring the l a r g e s t enerl3y)

.

The e a s i e s t would be i s o s p i n , l e f t - w v f n g z isospln: C IZiHiB

For e x q l e we would expect t h a t the

t h e @urnof each product hadron . s o e n s t o the

l e f t ( f o r e x a ~ l ei n the s y s t m with q pure @spacelike, o r myba the center of M -S), w i l l per c o l l i s i o n , f a r a g i v m x be given by

t h a t is, i t w i l l risae from zero f o r sm1l x (where u(x) r i s e sitrove

+ 112

( o r f a l l below

- 112)

etc,)

;(X)

and approach C 112 f o r

X

, never

near 1 (&ere,

we think u(x) d o ~ n a t e s ,) This r e l a t i o n can serve as can a n e e r of o t h e r s of t h i s type I n one of two ways,

In one case we ~ i g j n ttpuppose u(x), ;(X) e t c . , already knom frorn

eomc? other atethod a n s l y s f s of p f u r t h e r an.

-b

p

- such as neutrino -t

+ B B"" C anything,

scattering explained in l e c t u r e 33

- or

o r by perhapis one of t h e equsrtions developed

I n t h a t ease (36.1) is a q u a n t i t a t i v e prediction t o t e s t the

m s i s t e n c y of a l l the i d e m t h a t partons are! quarrks.

Alternatively

it

cm

i t s e l f be used t o give f u r t h e r i n f o m t i o n on t h e s i x separate functions u(x),

;(X)

etc., s o t h a t they m y be separately dete*ned,

Thieae could then

be cowared t o the r e s d t s of o t h e r raethods of ~ [ e t t i n gthem, but do n o t serve d i r e c t l y a s a check of the m d e l , Nonewr, even wlthout coaplete separation of a l l the functions a r e l a t i o n such a s (55.1) m y be used t o cheek t h e quark model because of the existence l l o f surrr m l e s (Equation 31.2) such a s (u(x) L(x))dx 2, (d(x) a(x))dx

-

-

so the integral, over x of the a m r a t o r of (56.1) should be 7/18.

I do not know a t t h i s tgne which kind of exper-ntal

infomtion w i l l

becorn available f i r s t and so my generat discussion s u f f e r s from a canfusfon Sn m a l y z i n g these t h e o r e t i c a l expectatlonra

a s t o which corns f i r s t the

m

1

horse o r c a r t .

There a r e a very l a r g e n&er

of such r e i a t i o n s whose

I w i l l therefore merely i n d i c a t e

organization is thereby m d e d i f f i c u l t .

some of t h e general r e l a t i o n s expected from our theofy and leave the choice of the b e s t way t o use them o r contbine them t o cowitre t o e x p e r i m a t up t o you,

Lecture 57

There is m o t h e r way t o insure t h a t m e g e t s a pure quark of one type r e c o i l i n g i n t h e f i n a l s t a t e , and t h a t is with deep i n e l a s t i c n e u t r i n o o r antiaeutrgnfno s c a t t e r i n g , a s we d i s m s s e d i n l e c t u r e 33.

We a s s m e of course,

weak coupling with t h e hadron p a r t of the weak current given t h e usual GJ*J U 1.l i n t e r n of q u a r b a s Cabibbo suggested, That is, we e x p l i c i t l y a s s m e :

We s h a l l a s e m f o r our d i s c w s i o n t h a t t h e current i n t e r a c t i o n is polnt-like, but t h a t i s a B a t t e r f o r e x p e r i m n t t o decide,

This very

i n t e r e s t i n g question i s , however, beside the scope of our discussion;

after

i t is d e t e d n e d t h e s a m kinds of questions and r e m a r h w i l l apply t o products generated by t h e aforementioned curneat operator.

I n aall cases

we can m a l y z e things a s i f they were the effect: of a v i r t u a l W meon f i e l d W of m a n t m q (generated by the lepton) coupled with t h e current: i n E4, li U J u s t as before (Equation 33.2) the t o t a l square rnatrix eleronent of J can Ir

be expressed i n t e r m of W2, W1, Wg s o t h e products, s u m d over spins and angles, a t l e a s t , c m a l s o be s o aslalyzed, each be s p l i t i n t o p a r t i a l Wl,

W2,

dlscuss oaXy t h e s c a l i n g region.

W

3

We have seen vW;! i o r e l a t e d t o W

- -

f i P - f y ( j u s t the s c a l e d function WV

WZ, WWg can

f o r s p e c i a l types of products.

r e l a t i o n should hold f a r products a b o ,

protons) was J u s t purely u(x).

That is t h e total, W1,

But =re

We

l and t h e

Lnteresting we noticed

ZHW3 f o r a n t i n e u t r i n o s c a t t e r i n g on

I-

Hence the s a m holds f o r t h e products, i e . ,

the p r o b a b i l i t y of a product i n M1, ( d e t e d n e d a s t h e appropriate c o e f f i c i e n t i n t h e v a r i a t i o n of cross s e c t i o n with laboratory n e u t r i n o angle klteepag q ,

V

fixed) minus t h a t i n W3 f o r ;p e c a t t e r i n g a r e purely the products f o r a u quark

being knocked backwards (Eomarda is a proton l e s s a d q w r k with p r o b a b i l i t y 2 cos BC

- the

.O6),

Thus by studying the products t o t h e l e f t i n t h i s cooibination we a r e

2 l a t t e r can i n f i r s t approxinration be negleczed a s s i n BC is only

studying the fragnaentacion products, Du(z), e~cpectedof one s i n g l e quark, a u quark i n f a c t . Ey choosing o t h e r ca&inations we can s e l e c t r e c o i l quarks of d i f f e r e n t types.

For example, f i P + f i p f o r neutrinos on protons gives pure

and should give various praducts with p r o b a b i l i t y ~;(z);(xl; t r i b u t i o n independent of

X,

the t o t a l cross s e c t i o n being

fiP-fgP f o r neutrinos on protons gives 2 d quark, s i n BC

a.

quarko

a f i x e d dis-

;(X),

Again

a pure d quark (cos

2

-94

.06 s quark),

From these d i s t r i b u t i o n s t h e t o t a l quantunt n m b e r s of the q m r k s can be deteracained. can n o t ,

Xn t h i s wag alone c e r t a i n m d e l s can be eliminated, but o t h e r s

For e x a w l e we cannot d i s c i n g d s h t h e quark model from t h e three-

t r i p l e t w d e l where t h e r e a r e 9 partons (and 9 antipartons) i n s e t s of three. A, B, and C each s e t having three s t a t e s l i k e quarks wi th various i n t e g e r

qwntum numbers,

Thus what would c o w out i n t h e experiment on fa-fg f o r

up s c a t t e r i n g , where we expect a pure u quark i n t h e q w r k rrrsdel, would, i n the t h r e e - t r i p l e t m d e l , be a -k l12 i s o s p l n parton b u t e i t h e r of type A o r

B o r C with equal p r o b a b i l i t y s o the mean charge on o t h e r n m b e r s can be l 1 3 i n t e g r a l and j u s t equal t o t h e u value, f o r t h a t is how t h e i n t e g e r charges f o r A, B, C were chosen.

(Other experiments, such a s e'e-

-t

anything o r

c o r r e l a t i o n s of l e f t and r i g h t s i d e d i s i n t e g r a t i a n s might d i s t i n g u i s h these models, )

There a r e a l a r g e nuniber of predictions i m p l i c i t i n the r e l a t i o n t h a t the d i s t r i b u t i o n of a given final hadron i n the l e f t (photon) d i r e c t i o n ( f o r convenience not n o m l i z e d ) is i n general given f o r deep i n e l a s t i c ep s c a t t e r i n g by: (Equation 55.1)

265

Bartans as Quarks where DU(z) e t c . , a r e t h e d i s t r i b u t i o n s of t h e product i n question f o r pure There a r e s i x f m c t i o n s i n general

up quarks, e t c .

80

analyze unless u(x) e t c . , were a l l alreadly a v a f l a b l e ,

i t is d i f f i c u l t t a

Howver, by taking

c e r t a i n co&inatfma of Eaewuremnts fewer functions axe involved,

%

i l l u s t r a t e t h i s with an e x a q l e ,

+

=4

Suppose we ask t o produce a etc.

WCg

c a l l t h e d i s t r i b u r i ~ n'D

By isospin r e f l e c t i o n t h e p r o b a b i l i t y a

t h a t a d produces a produces a

S-;

(z)

y i e l d s a n+ is t h e same a s

and by charge conjugation again t h e s a E a s t h a t a

In t h i s way

W-,

U

and :D

(x,z)

s e e f o r n production t h r e a r e r e a l l y only

WQ

t h r e e independent functions

In fact i f

kre

4"

measure t h e nu&er of n a n u s t h e a d e r of

R-

at a

given z i t (Equation 57.1) a l l reduces t o one function: 4"

D" (X,z>

- 'D

". ( x , ~ ) = A(z)

- V1 (d(x)-a(x)

(u(x)-;(x))

( f o r v i r t u a l y on proton) where A(z)

-

-k

:D

(x)

- :D

Thus we expect t h e d i s t r i b u t i o n ( p r o b a b i l i t y

(z).

as a function of z) t o be the same f o r a l l 4

(u(x)-;(X))

1

- $d(x)-8(x)),

X.

Aa we vary x we can deternine

within a constant.

This is j u s t a s i n 56.1, but

we do n o t have t o e a s u r e over a l l z t o i n t e g r a t e , and m a a u r e o t h e r p a r t i c l e s A =re

a s well,

wrzsurenaent of

3.

W

md

71-

a t soare convenient z would be enough.

The absolute c o e f f i c i e n t can be d e t e w n e d i n two ways, e i t h e r from t h e sum

-

r u l e s (Equations 31.21, o r by the hypothesis t h a t a s 4 a s ~ u ( x ) fep(x)

known function a s n

+

l.

X

-, 1 only u ( x ) s u m i v e s

Additional information would

came from t h e eame eltperiment on t h e neutmn, of course, (we g e t

g1

(U-;))

4

(dd)

+

e

The sum of the number of

W+

and

n- does not give

about the d i s t r i b u t i o n s , but we can roughly p r e d i c t its

US X

m& t h a t iet new dependence

The expression i n c u r l y brackets is t h e same a s fep(x) except f o r t h e c o e f f i c i e n t of t h e l a s t t e r n (which should be s l w l y 1 Is probably small ( f o r not only should

S

However, t h a t t e r n

,H be l e a s t h m say

i n a proton

U,:

4 r e l a t i v e t o $1 l but a l s o u is enhanced by F s o the d i s t r i b u t i o n of n' plus nis probably n e a r l y i n d e p e n d a t of x, and i f no-lized

t o fePfx) depend@ on

Arturo Gianeros haa suggested an, hypothesis which we explain i n w r e d e t a i l below, which a m u n t s t o a s e w n g t h a t near z

1 t h e f m c t i o n a ;D,(z)

f a l l o f f with variouhl powers of (I-z) and i n p a r t i c u l a r chat a s z

+

+

function :D

i s much l a r g e r than

+

o r :D

:D

.

-t.

l the

This .lakes t h e c o e f f i c i e n t s

of (57,3), (57.4) equal. a s z -, l, Thus i t m a n s t h a t ia t h i s region t h e p r o b a b i l i t y of f i n d i n g a n' a s we vary x is a d i r e c t measure of ufx) and t h e p r o b a b i l i t y of fgnding a n- m a s u r e e ;(X)

+

+ a(.),

d(x) i n t h e s a a e e a l e ,

This i s s t i l l another suggestion of how the i n d i v i d u a l determination of the f m c t i o n s na(x) m y be f a c i l i t a t e d . pe-ts

Xn f a c t , t h i s hypothesis, i f t r u e ,

a d e t e d n a t t o n of the s i x functions u(x), 5(x) e t c . (up t o an

o v e r a l l a u m r i c a l canstant) , by memuring the dia t r i b u t i o n functions f o r charged m s o n s only near z proton i s

l, f o r both proton asld neutron.

used a s a t a r g e t t h e n

(X)

Sf only t h e

cannot be d a t e m i n e d without =king

.trzemuremnts o f neutral. plebsons which is d i f f i c u l t experinrttntallg, Me can do eintilar things f o r t h e production of o t h e r p a r t i c l e s , f o r example K-mesons

.

Here t h e r e a r e s i x independent functions Df(z),

o t h e r s a r e obtained by f s o s p i n r e f l e c t i o n o r charge conjugation.

D 5 $ - 6' -

-d

D$

.

the (For e x m l e ,

The student e m v e r i f y t h a t f o r u on protons

+(2)

i f we m a s u r e p a r t i c l e s a t a given z t o t h e l e f t , i f N 4-

K e t c . , we f i n d t h e following r e s u l t s : on one f w c t i o n , with t h e s a w

m o t h e r coIlrbination (*ich

X

i s t h e n m b e r af

The i s o t o p i c s p i n difference depends

dependent c o e f f i c i e n t as before:

does not require R'

and

a l s o f a c t o r i z e s i n t o one f w c t i o n OE x , one of z:

p to

be distinguished)

P a r t m as Quarks

Ttre suxtl of a l l four depend@ s t r i c t l y on two functions o f z :

but the last f a c t o r i n curly brackets is l i k e l y t o be close t o fepfx) f o r any z , F i n a l l y t h e fourth r e l a t i o n , illrrolving two funetions is

(a measure of hyperckarge, but unequally s e n s i t i v e t o stxmge- a d non-stirange

quarks s i n c e we do a o t a s s m e SUj invariance). Cisneros' aassmption E(i(be1ow) h e r e a m @t h a t , a s z survive, c a l l them a , @ respectively. as z

+

-, 1,

4-

3.

only DU a d D@

Then a rseasure of K mesons t o t h e l e f t

1 is a dSLrect marsure of v a r i o m c d i n a t i o r a s of the u,d functions.

In t h i s , a s i n a l l cases, data on vn gives a d d i t i o n a l infornation, change ufx)

d(x) and

;(X)

*-+

Z(x) fi t h e f o m u l a a ,

Me have n o t discussed t h e r i g h t d i s t r i b u t i o n s , but t h e r e a r e r e l a t i o n s h e r e too f o r various e x p e r i m n t s ,

I?ftz mgneion only one as an e x a q l e , ""

Llewellyn Smith' s sum r u l e (Equation 33.6) neglecting s i n2 BC f;p(x)-f;p 4(ap{x)

- fen(x))

works f o r every x a a a t o t a l cross seetlton,

-

k now

s e e t h a t t h e o b j e c t s produced a t t h e right ( i n the hadran f r a v n t e t i c m region) a r c the s a w on both a i d e s f o r t h e proton but n o t f o r t h e neutron f o r &a en experiment we nrwt observe the ieoepin r e f l e c t e d products.

-

If

t h i s is done, the r e l a t i o n holds f o r t h e p a r t i a l cross s e c t i o n s f a r any products t o the ri&t i f the l e f t producta a r e n o t observed. Finally

CO

b r i n g a l l t h e hypotheses we have- made &out pertons together

i s t o one l i s t we note f i n a l l y our suggestion t h a t the $"ifep suggests t h a t when a protan h a s a quark near x

mmatuol, t h a t quark Is a

U

quark.

3

r a t e s strongly

1 and a remainder of snslllf,

We t r y t o genara2ltze t h i s t o any barnon

of t h e 56 nrultiplet i n $U6 language a m i n g our $U6 only quatlitativaly, n a t e z a ~ t l yq w n e i t a e l v e l y .

Photon-Hadran Interaetiom

268

A %&er

of t h e deciroet: 4, h a s a s w f l e r probability of having a quark

near x = 1 than does t h e o c t e t . f a r t o t a l c r o s s sections,

We have already discussed t h e i a p l i c a t i o n s

X t has i q l i c a t i o n s f o r products a l s o , of course.

klcz a l s o a s s u m t h a t i f a s t a t e i s a pure quark t o t h e l e f t i t has an

m l i t l x d e t o be a b a q o n t o the l e f t with z n e a r l y 1 which is proportional the the chance t h e baryan contains a f a s t quark near m e r e a r e them w a y implleations Eor products

X

= Z of the s a w kind,

- and we have ~ t i o n e dso=

t h a t came from u(x) being 1arlger than a l l t h e o t h e r s a s x

-+

I,

There a r e

o t h e r s , of another type, f o r e x a q l e (Claneroa, p r i v a t e comunication) i n eke 4"

e e

-

-c

hrrdrons s i n e e 5u haa four times t h e p r o b a b i l i t y of ad the chance o f

producing a proton with x n e a r l ( i n t h e c e n t e r of =ss)

and anything, e l s e is

f o u r times the probabi l i t y of producing a neutron, I f &S 5s c o r r e c t we shauld l i k e t o a s s u w solnething analogous f o r t h e The analogoos a s s m p t i o n Fa t h a t when one quark takes m e t of the

=ems.

mwatufn i t i s of a type t h a t t h e low e n e r m quark =del t o be m b e o f .

Cbe t h e r e f o r e (Zn agreemeat with Zisneros) aserne:

( f o r t h e charge conjugate i t i a it8

Q

supposes t h e meon

@

near x = l and the remainder

.

We have b u i l t a very t a l l house of cards asking s o m a y weakly-based eonJectures one upon t h e o t h e r and a g r e a t d e a l anay be wrong.

(Probably

- s a m p l a t e a u f o r gap and hadron - b u t i f i t were wrong i t does not a l t e r t h e t h r w t of any of t h e o t h e r s - j u s t i n t h e operator expreesions the weakest is C6

we s h a l l have t o be c a r e f u l t o u l ~ et h e rgght plateauh),] tdevertheless t h i a is the b e s t guess I can riitaake

ROW

- and we- can t r y t o use them has warking hypotheses.

Probably t h e g r e a t e s t challenge t o e x p e r i w n t and theory is t o get soracr, evidence o f quark qumurturn n u d e r s i n high-energy collisions,

The low-energy

quark ausdel, good a s i t is, i a n o t enough, t h e r e i s always l i n g e r i n g doubt that: the r e g u l a r i t i e s observed have so=

e n t i r e l y d i f f e r e n t bask8 o r arc?,

i n part, accidental,

The establSshment of evidence f o r the quark model

( m d we have i n d i c a t e d very nrany ways

- both

i n the l a s t few l e c t u r e s a s w e l l

a s e a r l i e r -(Llewellyn S ~ t ' sh sum r u l e (Equation 33.61, t h e s p i n sum r u l e f o r glp-gIn

e t c . ) by high energy e n p e r i m n t s would c o n f i w a t once the r e a l i t y

of the r e g u l a r i t i e s i n t e r p r e t e d by the low energy quark aaotlel.

This would

m k e f i r m a e m j e c t u r e of deepest s i g n i f i c a n c e t o understanding high energy physics

-

t h e intgortance of quark q u a a t m n m b e r s .

Supposing f o r a m m n t t h i s i s done, t h a t h e next s e r i o u s question w i l l become t h e o r e t i c a l

- what

e x a c t l y is t h e r e l a t i o n of t h e quark q u a l i t i e s a t

high energy and, t h a s e a t low e n e r m .

"I"heW"prtans a s q u a r b " "del

does not

iarply t h e low energy model ( i . e . why a r e the wave f m c t i o n s n o t m r e c o q l i c a t e d involving quark antiquark p a i r s ) n o r v i c e versa. would n o t b e understood. courage

- you mi&t

confirmed,

A t present t h e i r relation

To s t a r t working on t h i s n m w i l l take a l i t t l e

waste your time

- mybe

partons a s quarks w i l l not be

I f you do s t a r t , p o s s i b l y one place t o s t a r t might be t o t h i n k

about low energy m a t r i x eferoents l i k e d

-c

p

+y

i n a fahit m v i n g systetn i n

which a l l ( o r some) m m n t a a r e of t h e o r d e r P s o parton wave f m c t i o n s can be used,

(We have one r e l a t i o n of t h i s kind i n Bjorken"

sslun

r u l e f a r g,

Equation 3 3 - 1 6 . ) F i n a l l y i t should be noted t h a t even i f our house of cards s u r v i v e s and proves t o be r i g h t , we have n o t thereby proved t h e e x i s t e n c e a f partons.

'Ilhe

final, r e s u l t of o u r c o n s i d e r a t i o n s has been t o d a a c r i b e ehe r e s u l t of t h e

-

operation of a c u r r e n t on a proton s t a t e .I /p> ( f o r l a r g e v , -q2 ZMur) a s U a l i n e a r corobination of o p e r a t o r s l i k e D ~ E ~ ( ~ - ~ , ~ ~ M /c Yr eAact i,n g f i n a l outgoing hadron s t a t e s only,

It might be wise t o follow t h i s o u t f o m a l l y

without mentioning partons (analogous t o t h e way Gel1-Mann and F r i t z s c h d e s c r i b e parton r e s u l t s f o r t o t a l i n c l u s i v e c r o s s s e c t i o n s i n tevma of c a r n u t a t i o n r u l e s f o r q u a n t i t i e s , c u r r e n t s , defined i n general whether partans

""exist" o r r o t ) . From t h f s p o i n t a f view t h e partons would appear as an unnecessary s c a f f o l d i n g t h a t was used i n b u i l d i n g our house of cards. On t h e o t h e r hand, t h e partons would have been a u s e f u l psychological guide a s t o what r e l a t i o n s t a expect

- and

i f they continued t o s e r v e t h i s

270

PIZO~OB-R~TOP~ Intemctioll~

way to produce other valjid expectations they would of course b e d n to becclr~e "'real'"

pposiibly as real as any other thczoretjlcal structure invented to

bscrjlbe lature. A t any rate we shall see.

to,

It is good to have sowthing to look forvard

Appendix A. The sospin of Quark Fragmentatior Products

The dfseussion (Lecture 56) Leading t o the idea t h a t a d d i t i v e quark quantum n d e r s could appear a s mean t o t a l quantum n d e r s of p r ~ d u c t sm v i n g i n one d i r e c t i o n i s s u r p r i s i n g

- e s p e c i a l l y when i t

is

noted t h a t what holds f o r 3-isospin holds a l s o f o r any o t h e r c o q o n e a t such a s l-isospin o r 2-isospin (although of course i n p r a c t i c e they a r e nearly inrpossible t o measure).

It looks l i k e m isaospin 112 o b j e c t

could be represented by a group a f i s o s p i n l obJects (e.g. pions) which a t f i r s t s e e m irapossible

- except

-

t h a t we have an i n d e f i n i t e

n m b e r of such o b j e c t s , It i s t h e r e f o r e of i n t e r e s t t o make a very simgle s p e c i a l =the-

m t i c a l -del,

t o show t h a t indeed such things can be done i n p r i n c i p l e .

This i s e s p e c i a l l y i a p a r t a n t when i t is r e a l i z e d t h a t our previous attempts a t m a t h e m t i c a l f o m u l a t i o n cannot be copletcr; and m w t be looked a t a s mere m e m n i c s (see note i n Iazeture SS on the D and E operators).

This l i t t l e ntodel may h e l p by its exalngle t o l e a d t o c o r r e c t

pososfble formal expressians of our ideas. I n t h i s model suppose quarks c a r r y only fsospfn 112 m d hadrons a r e

271

only pions of isospin 1

- made

of quark antiquark p a i r s .

Zmgine t h a t we s t a r t with some current a n n i h i l a t i o n (analogous t o eie- but i n more general. isospin d i r e c t i o n ) i n i t i a l l y d i s i n t e g r a t i n g i n t o a p a i r of quarks Qa,

GB

(a,@ a r e SU2 i n d i c e s fixed i n t h i s problem.

I a e d i a t e l y a f t e r interaction:

-P

P

Q@

N quark p a i r s i n s i n g l e t s t a t e

a f t e r M m i l t m i a n operates;

m k e s hirdrons

o f type

Next the a c t i o n of the Wamiltonian f s t o produce p a i r s of quarks i n a s i n g l e t s t a t e uniformly spaced i n y space

- a t y p i c a l one i s qihi

The number o f such p a i r s M is then proportional

d on a l l I, equally.

t o Rn2P whlch we take t o be very l a r g e ,

( h e could a l s o ass-

the

n u d e l : d i s t r i b u t e d v i a Poisson with a man M e t c . , but we avoid compli-

- choose N

cations which only serve t o confuse our point

fixed.)

Next t h i s row of: quarks is a a s m d t a convert t o pions by a simple Thus ( i n

r u l e , each pion is f o ~ by d a p a i r adjacent i n t h e y space. figure) I f the f i r s t new s h g l e t p a i r had index 1, the next f i r et n

3 e t c . the

f o m d front an antiqmrtr index fl and a quark, index f ;

igi

next by an antiquark index

1,

and a qusrk index

3;

- the

- etc. ?.

To describe the i s o s p i n a t a t e of a n we we an isospin 3-vector

+

Thus i f the s is a n e u t r a l pion wo we have V with only a z cowoneat,

(0,0,1).

4-

For a a we have

1 ?=(l,f,O) 42-

etc.

antiquark of index y and a quark of index 6 fom a vector

if

The amplitude t h a t an 7~

characterized by

i s then proportianal t a the yd m t r i x element of the two-

by-rwo ~ o a t r i xoeif where

U

a r e the Pauli m t r i c e s ,

m i t e t h i s a s .

(WE! work i n r e l a t i v e m p l i t u d e e and p r o b a b i l i t i e s l a v i n g overall.

n o m l i z a t i m r o the end. )

Bus t h e t o t a l amplitude t o f i n d t h e n ' s i n d i r e c t i o n s ?l$2. ..qH i s Amp =

2

........ ~ * ? ~ l o,>the

< ~ / o l . ? ~ 1 ie>i j l ~ = 5 ~ 1,

For s t a t e

p@

/o%